DP Maths AA HL - Calculus - Integration and Its Applications

DP Maths AA HL - Calculus - Integration and Its Applications - Full notes

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DP Maths AA HL | Integration and Its Applications

Integration and Its Applications

This topic contains 2 notes:

5.5 Integration
5.16 Further Integration

DP MATHEMATICS AA · TOPIC 5: CALCULUS

5.5

Integration

Full notes for HL · Syllabus SL 5.5, 5.10, 5.11

Contents

1	Antidifferentiation
2	Standard integrals
3	Finding the constant of integration
4	Reverse chain rule and substitution
5	Definite integrals
6	Area under a curve
7	Area between two curves
8	Kinematics with integration
9	Indefinite integrals: more practice
10	Integrals of linear compositions
11	Integration by inspection (reverse chain rule)
12	Definite integrals with exact answers
13	Properties of definite integrals
14	Areas under curves: more practice
15	Areas between curves: more practice
16	Kinematics with integration
17	Accumulation problems
18	Trapezoidal-style estimates and technology
19	Mixed review: worked problems
20	Additional exam-style practice
21	Integration and differentiation together
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
understand integration as the reverse of differentiation
integrate xn (n ≠ −1), 1/x, ex, sin x and cos x, and linear compositions of these
find the constant of integration from a boundary condition
integrate by inspection (reverse chain rule) and by simple substitution
evaluate definite integrals exactly and with technology
find areas under curves and between curves, including regions below the x-axis
find displacement and distance travelled from velocity

This booklet covers SL 5.5, 5.10, 5.11, which is common to SL and HL. HL students extend this topic in booklet 5.16.

Guiding questions

How does reversing differentiation lead to areas? How can we recover position from velocity?

How this topic is examined

Integration appears on every paper: indefinite integrals (including reverse chain rule), finding f from f′ and a point, exact definite integrals on Paper 1, areas between curves with technology on Paper 2, and kinematics (displacement and total distance).

Always include + c for indefinite integrals, and split areas at the x-axis when the curve crosses it.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 Antidifferentiation

Undoing the derivative.

Think about it

If the gradient of a curve is 2x everywhere, what could the curve be? Is there only one possible answer?

Integration reverses differentiation. If F′(x) = f(x), then F is an antiderivative of f, and we write the indefinite integral:

The constant of integration c appears because the derivative of a constant is zero: x2, x2 + 5 and x2 − 3 all have derivative 2x. The antiderivatives of a function form a family of curves, vertical translations of each other.

Figure 1. The family of antiderivatives of 2x.

2 Standard integrals

Reverse each derivative.

f(x)	∫f(x) dx
xn (n ≠ −1)	xn+1/(n + 1) + c
1/x	ln\|x\| + c
ex	ex + c
sin x	−cos x + c
cos x	sin x + c
(ax + b)n	(ax + b)n+1/(a(n + 1)) + c
eax+b	eax+b/a + c
cos(ax + b)	sin(ax + b)/a + c

To integrate xn, add 1 to the power and divide by the new power. Rewrite roots and fractions as powers first, and expand products. For a linear inside function ax + b, integrate as usual and divide by a.

WORKED EXAMPLE 1 Powers (basic)

Find ∫(6x2 − 4/x3 + 3√x) dx.

Solution

= 6x3/3 − 4x−2/(−2) + 3x3/2/(3/2)M1

= 2x3 + 2/x2 + 2x3/2 + cA1 A1

WORKED EXAMPLE 2 Linear compositions (standard)

Find (a) ∫e3x−1 dx; (b) ∫sin(2x) dx; (c) ∫1/(2x + 5) dx.

Solution

(a) e3x−1/3 + cA1

(b) −cos(2x)/2 + cA1

Common mistake

Integrating 1/x as x0/0. The power rule fails for n = −1; the integral is ln|x|.

Practice exercise 1

1. Find ∫(x3 − 2x + 7) dx.

2. Find ∫(x + 1)2/x2 dx (expand first).

3. Find ∫(4e2x − cos 3x) dx.

4. Find ∫(3x − 2)5 dx.

3 Finding the constant of integration

Use a known point.

WORKED EXAMPLE 3 Curve through a point (standard)

A curve has gradient dy/dx = 3x2 − 4x and passes through (2, 5). Find its equation.

Solution

y = x3 − 2x2 + cA1

5 = 8 − 8 + c, c = 5M1

y = x3 − 2x2 + 5A1

Practice exercise 2

1. f′(x) = ex + 1 and f(0) = 4. Find f(x).

2. f″(x) = 6x, f′(1) = 0 and f(1) = 2. Find f(x).

4 Reverse chain rule and substitution

Spot the derivative of the inside.

If an integrand has the form g′(x) f(g(x)), the integral is F(g(x)) + c, where F is an antiderivative of f. In practice: look for a function and (a multiple of) its derivative. Examples: ∫2x(x2 + 1)4 dx = (x2 + 1)5/5 + c; ∫g′(x)/g(x) dx = ln|g(x)| + c.

Substitution makes this systematic: let u = g(x), so du = g′(x) dx; rewrite the integral in u, integrate, then substitute back.

WORKED EXAMPLE 4 By inspection (standard)

Find ∫x2ex³ dx.

Solution

Inside x3 has derivative 3x2M1

= ex³/3 + cA1

WORKED EXAMPLE 5 By substitution (standard)

Find ∫x√(x2 + 9) dx using u = x2 + 9.

Solution

du = 2x dx, so x dx = du/2M1

∫½u1/2 du = u3/2/3A1

= (x2 + 9)3/2/3 + cA1

WORKED EXAMPLE 6 A log integral (exam level)

Find ∫(6x + 3)/(x2 + x + 1) dx.

Solution

Numerator = 3 × derivative of the denominatorM1

= 3 ln(x2 + x + 1) + cA1

Practice exercise 3

1. Find ∫cos x sin3x dx.

2. Find ∫x/(x2 + 4) dx.

3. Find ∫(2x + 1)ex² + x dx.

5 Definite integrals

Evaluate between limits.

This is the fundamental theorem of calculus. No + c is needed, because it cancels. Properties: ∫ from a to a = 0; reversing the limits changes the sign; ∫ from a to b + ∫ from b to c = ∫ from a to c; constants can be taken outside.

WORKED EXAMPLE 7 Exact definite integral (standard)

Find the exact value of ∫ from 0 to π/2 of (sin x + 2x) dx.

Solution

[−cos x + x2] from 0 to π/2M1

= (0 + π2/4) − (−1 + 0)A1

= π2/4 + 1A1

WORKED EXAMPLE 8 Exact log answer (exam level)

Find ∫ from 1 to 3 of 2/(2x + 1) dx exactly.

Solution

[ln|2x + 1|] from 1 to 3M1

= ln 7 − ln 3 = ln(7/3)A1

GDC tip

Use the numerical integration function (∫ dx) for definite integrals on Paper 2. For areas, integrate the absolute value, or split at the roots.

Practice exercise 4

1. Evaluate ∫ from 1 to 4 of √x dx.

2. Evaluate ∫ from 0 to 1 of e2x dx exactly.

3. Given ∫ from 1 to 3 of f(x) dx = 5, find ∫ from 1 to 3 of (2f(x) + 1) dx.

6 Area under a curve

Split at the x-axis.

For f(x) ≥ 0 on [a, b], the area between the curve and the x-axis is ∫ from a to b of f(x) dx. If part of the curve is below the axis, the integral of that part is negative. To find total area, split at the roots and add the absolute values (or integrate |f(x)| with technology).

WORKED EXAMPLE 9 Area crossing the axis (exam level)

Find the total area enclosed between y = x2 − 4 and the x-axis for 0 ≤ x ≤ 3.

Solution

Root at x = 2; below axis on [0, 2]M1

∫ from 0 to 2 = [x3/3 − 4x] = 8/3 − 8 = −16/3; ∫ from 2 to 3 = (9 − 12) − (8/3 − 8) = 7/3A1 A1

Area = 16/3 + 7/3 = 23/3A1

Figure 2. The region between y = x2 − 4 and the x-axis, split at x = 2.

Common mistake

Integrating straight from 0 to 3 gives −3, not the area 23/3, because the negative part cancels some of the positive part.

Practice exercise 5

1. Find the area under y = 6x − x2 between x = 0 and x = 6.

2. Find the area enclosed by y = sin x and the x-axis for 0 ≤ x ≤ 2π.

7 Area between two curves

Top minus bottom.

Find the intersection points (the limits), decide which curve is on top, and integrate top minus bottom. If the curves cross within the interval, split at the crossing points.

WORKED EXAMPLE 10 Area between curves (standard)

Find the area enclosed by y = x2 and y = x + 2.

Solution

x2 = x + 2: x = −1, 2A1

∫ from −1 to 2 of (x + 2 − x2) dx = [x2/2 + 2x − x3/3]M1

= (2 + 4 − 8/3) − (1/2 − 2 + 1/3) = 9/2A1

Figure 3. The region enclosed by y = x2 and y = x + 2.

Practice exercise 6

1. Find the area between y = x2 and y = 2x.

2. Use technology to find the area enclosed by y = ex and y = x + 3.

8 Kinematics with integration

From acceleration to velocity to displacement.

Integrating velocity gives displacement; the constant is found from the initial position. The definite integral ∫ from t1 to t2 of v dt gives the displacement over that interval, while ∫|v| dt gives the total distance travelled.

WORKED EXAMPLE 11 Distance and displacement (exam level)

A particle has v(t) = t2 − 4t + 3 m/s. Find its displacement and the distance travelled for 0 ≤ t ≤ 3.

Solution

Displacement = ∫ from 0 to 3 of v dt = [t3/3 − 2t2 + 3t] = 9 − 18 + 9 = 0 mM1 A1

v changes sign at t = 1 and t = 3: ∫ from 0 to 1 = 4/3, ∫ from 1 to 3 = −4/3M1

Distance = 4/3 + 4/3 = 8/3 mA1

Practice exercise 7

1. a(t) = 6t − 4, v(0) = 2, s(0) = 0. Find s(t).

2. Use technology to find the distance travelled in the first 5 seconds by a particle with v(t) = 3 cos t.

9 Indefinite integrals: more practice

Rewrite, then integrate.

Expand products, split fractions with a single-term denominator, and write roots as powers before integrating. Always include + c.

WORKED EXAMPLE 12 Fraction (standard)

Find ∫(x3 − 2x + 5)/x2 dx.

Solution

∫(x − 2/x + 5x−2) dxM1

= x2/2 − 2 ln|x| − 5/x + cA1 A1

WORKED EXAMPLE 13 Roots (standard)

Find ∫(√x + 1/√x) dx.

Solution

∫(x1/2 + x−1/2) dxM1

= (2/3)x3/2 + 2x1/2 + cA1

Practice exercise 8

1. Find ∫(2x + 1)2 dx.

2. Find ∫(4 − x)/√x dx.

3. Find ∫(ex + 1)2 dx.

10 Integrals of linear compositions

Divide by the coefficient of x.

∫f(ax + b) dx = F(ax + b)/a + c. This covers (ax + b)n, eax+b, sin(ax + b), cos(ax + b) and 1/(ax + b).

WORKED EXAMPLE 14 Linear compositions (standard)

Find (a) ∫(5 − 2x)4 dx; (b) ∫cos(3x − 1) dx; (c) ∫4/(2x + 3) dx.

Solution

(a) −(5 − 2x)5/10 + cA1

(b) sin(3x − 1)/3 + cA1

Practice exercise 9

1. Find ∫e1−3x dx.

2. Find ∫sin(πx/2) dx.

11 Integration by inspection (reverse chain rule)

Spot f′(x) multiplying f(x).

If an integrand is (a multiple of) g′(x) times a function of g(x), integrate the outer function and substitute back. Check by differentiating.

WORKED EXAMPLE 15 Inspection (exam level)

Find ∫x(x2 + 3)5 dx and ∫sin x cos4x dx.

Solution

(x2 + 3)6/12 + cA1

−cos5x/5 + cA1

Practice exercise 10

1. Find ∫3x2ex³ dx.

2. Find ∫(2x + 1)/(x2 + x + 5) dx.

12 Definite integrals with exact answers

Evaluate carefully.

Evaluate F(b) − F(a), keeping exact forms (fractions, π, ln, e). Bracket negative terms when subtracting.

WORKED EXAMPLE 16 Exact (standard)

Evaluate ∫ from 0 to π/3 of (2 sin x + cos 2x) dx.

Solution

[−2 cos x + ½ sin 2x] from 0 to π/3M1

= (−1 + √3/4) − (−2)A1

= 1 + √3/4A1

Practice exercise 11

1. Evaluate ∫ from 1 to e of (1 + 1/x) dx.

2. Evaluate ∫ from 0 to ln 2 of e2x dx.

13 Properties of definite integrals

Splitting and reversing.

∫ from a to b of kf = k∫ from a to b of f; ∫ from a to b + ∫ from b to c = ∫ from a to c; ∫ from b to a = −∫ from a to b. These let you use given integral values to find others.

WORKED EXAMPLE 17 Using properties (exam level)

∫ from 1 to 4 of f(x) dx = 7 and ∫ from 1 to 6 of f(x) dx = 10. Find ∫ from 4 to 6 of (3f(x) + 2) dx.

Solution

∫ from 4 to 6 of f = 3A1

3(3) + 2(2)M1

= 13A1

Practice exercise 12

1. Given ∫ from 0 to 5 of g(x) dx = 8, find ∫ from 5 to 0 of 2g(x) dx.

14 Areas under curves: more practice

Positive and negative regions.

Total area requires splitting at the roots and taking absolute values; a GDC can integrate |f(x)| directly.

WORKED EXAMPLE 18 Cubic (exam level)

Find the total area between y = x3 − 4x and the x-axis.

Solution

Roots −2, 0, 2; odd functionM1

∫ from −2 to 0 = 4A1

Total area 8A1

Practice exercise 13

1. Find the area enclosed by y = sin x and the x-axis for 0 ≤ x ≤ 3π/2.

2. Find the total area between y = x2 − x − 2 and the x-axis for 0 ≤ x ≤ 3.

15 Areas between curves: more practice

Upper minus lower.

Find intersections, identify which function is above, and integrate the difference. When curves cross more than once inside the interval, split at each crossing.

Figure 4. Two regions between y = x3 and y = x.

WORKED EXAMPLE 19 Two regions (exam level)

Find the total area enclosed by y = x3 and y = x.

Solution

Intersections x = −1, 0, 1A1

∫ from 0 to 1 of (x − x3) dx = 1/4M1 A1

Total 1/2 (by symmetry)A1

Practice exercise 14

1. Find the area between y = ex and y = e−x for 0 ≤ x ≤ 1.

2. Find the area between y = 4 − x2 and y = x + 2.

16 Kinematics with integration

From acceleration to displacement.

v = ∫a dt and s = ∫v dt, using initial conditions to find constants. Displacement is ∫v dt; distance is ∫|v| dt.

WORKED EXAMPLE 20 Acceleration (exam level)

a(t) = 6 − 2t, v(0) = 4, s(0) = 0. Find v(t), s(t) and the time the particle is at rest.

Solution

v = 6t − t2 + 4A1

s = 3t2 − t3/3 + 4tA1

v = 0: t2 − 6t − 4 = 0: t = 6.61M1 A1

Practice exercise 15

1. Find the distance travelled in the first 8 seconds.

17 Accumulation problems

Totals from rates.

If r(t) is a rate (litres per minute, people per hour), then ∫ from a to b of r(t) dt is the total change over [a, b]. Add the initial amount to find the final amount.

WORKED EXAMPLE 21 Water tank (standard)

Water enters a tank at r(t) = 30 − 0.5t L/min for 0 ≤ t ≤ 40, starting with 200 L. Find the volume at t = 40.

Solution

∫ from 0 to 40 = 1200 − 400 = 800M1 A1

Volume 1000 LA1

Practice exercise 16

1. A town's population grows at 120e0.02t people per year. Find the increase over the first 10 years.

18 Trapezoidal-style estimates and technology

When no antiderivative is known.

Some integrals, such as ∫e−x² dx, have no elementary antiderivative. Evaluate definite integrals numerically with the GDC; for data, a sum of trapezia estimates the area.

WORKED EXAMPLE 22 Numerical (standard)

Evaluate ∫ from 0 to 1 of e−x² dx.

Solution

GDC: 0.747A1

Practice exercise 17

1. Evaluate ∫ from 0 to π of sin(x2) dx with technology.

19 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 23 Mixed problem (exam level)

f′(x) = 6x2 − 4x and f(1) = 5. (a) Find f(x). (b) Find the area between y = f′(x) and the x-axis for 0 ≤ x ≤ 1.

Solution

(a) f = 2x3 − 2x2 + 5A1 A1

(b) f′ < 0 on (0, 2/3): splitM1

∫ from 0 to 2/3 = −8/27 and ∫ from 2/3 to 1 = 8/27: total area 16/27A1

Practice exercise 18

1. Find k > 0 such that ∫ from 0 to k of (2x + 1) dx = 12.

20 Additional exam-style practice

Paper 1 and Paper 2.

WORKED EXAMPLE 24 Paper 1 question (exam level)

(a) Find ∫x√(x2 + 1) dx. (b) Hence find the exact value of ∫ from 0 to √3 of x√(x2 + 1) dx.

Solution

(a) (x2 + 1)3/2/3 + cM1 A1

(b) (8 − 1)/3 = 7/3A1

Practice exercise 19

1. Find the area enclosed by y = 2x and y = x2 − 3.

21 Integration and differentiation together

The fundamental theorem.

d/dx ∫ from a to x of f(t) dt = f(x), and ∫ from a to b of f′(x) dx = f(b) − f(a). These link the two halves of calculus and allow quick evaluation when f is known.

WORKED EXAMPLE 25 FTC (standard)

Find ∫ from 0 to 2 of d/dx(x2ex) dx.

Solution

= [x2ex] from 0 to 2M1

= 4e2A1

Practice exercise 20

1. Find d/dx of ∫ from 1 to x of ln t dt.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 15 minutes)

Let f(x) = 3x2 − 6x and g(x) = 6 − 3x.

(a) Find the x-coordinates of the points of intersection of the graphs. [3]

(b) Find the area of the region enclosed by the two graphs. [4]

Markscheme

Part	Answer	Marks
(a)	3x2 − 6x = 6 − 3x	M1
	3x2 − 3x − 6 = 0, x2 − x − 2 = 0	A1
	x = −1, 2	A1
(b)	∫ from −1 to 2 of (6 − 3x − 3x2 + 6x) dx	M1
	[6x + 3x2/2 − x3] from −1 to 2	A1
	(12 + 6 − 8) − (−6 + 3/2 + 1)	A1
	13.5	A1

Examiner's comments. Identifying g as the upper function between the roots is essential.

Question 2 (Paper 2 style) (about 10 minutes)

A car accelerates from rest with v(t) = 10(1 − e−0.2t) m/s.

(a) Find the acceleration when t = 5. [2]

(b) Find the distance travelled in the first 10 seconds. [2]

Markscheme

Part	Answer	Marks
(a)	a = 2e−0.2t	M1
	0.736 m/s2	A1
(b)	∫ from 0 to 10 of v dt	M1
	56.8 m	A1
(c)	approaches 10 m/s	A1

Examiner's comments. Technology is expected for (b).

23 Common misconceptions and links

Students often think…	Actually…
forgetting + c	include it for indefinite integrals
∫1/x dx = x0/0	ln\|x\|
area = integral even below the axis	split at roots
displacement = distance	distance uses \|v\|

How 5.5 connects to the rest of the course

Topic	Connection
5.1 Differentiation	reverse process
5.8 Kinematics	velocity and acceleration
4.14 Continuous RVs (HL)	probability as area
5.16 Further integration (HL)	parts, volumes

24 Check your understanding

Section A: multiple choice

1. ∫x4 dx =

A. x5/5 + c B. 4x3 C. x5 + c D. 5x5

2. ∫e2x dx =

A. 2e2x + c B. e2x + c C. e2x/2 + c D. ex²

3. ∫cos x dx =

A. −sin x + c B. sin x + c C. cos x + c D. −cos x + c

4. ∫ from 0 to 2 of 3x2 dx =

A. 12 B. 6 C. 4 D. 8

5. Distance travelled is

A. ∫v dt B. ∫|v| dt C. ∫a dt D. |∫v dt|

Section B: structured

6. Find the area between y = 4 − x2 and the x-axis. [3]

7. Find f(x) if f′(x) = 1/x and f(1) = 3. [2]

Answers

1. A 2. C 3. B 4. D 5. B

6. 32/3.

7. ln x + 3.

25 Summary and topic map

Idea	Remember
Antiderivative	∫f dx = F + c
Power rule	xn+1/(n + 1), n ≠ −1
Standard	1/x → ln\|x\|; ex; sin → −cos; cos → sin
Linear inside	divide by a
Reverse chain	g′(x)f(g(x))
Definite	F(b) − F(a)
Areas	split at roots; top minus bottom
Kinematics	s = ∫v dt; distance ∫\|v\| dt

Before the exam, make sure you…

•include + c
•rewrite as powers
•split areas at the axis
•use |v| for distance

Integration on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 5. Topic map for Integration.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. x4/4 − x2 + 7x + c

2. x + 2 ln|x| − 1/x + c

3. 2e2x − (1/3)sin 3x + c

4. (3x − 2)6/18 + c

Practice exercise 2

1. ex + x + 3

2. f′ = 3x2 − 3, f = x3 − 3x + 4

Practice exercise 3

1. sin4x/4 + c

2. ½ ln(x2 + 4) + c

3. ex² + x + c

Practice exercise 4

1. 14/3

2. (e2 − 1)/2

3. 12

Practice exercise 5

1. 36

2. 4

Practice exercise 6

1. 4/3

2. intersections −2.95 and 1.51: area 5.69

Practice exercise 7

1. v = 3t2 − 4t + 2; s = t3 − 2t2 + 2t

2. ∫ from 0 to 5 of |3 cos t| dt = 9.12 m

Practice exercise 8

1. (4/3)x3 + 2x2 + x + c

2. 8√x − (2/3)x3/2 + c

3. e2x/2 + 2ex + x + c

Practice exercise 9

1. −e1−3x/3 + c

2. −(2/π)cos(πx/2) + c

Practice exercise 10

1. ex³ + c

2. ln(x2 + x + 5) + c

Practice exercise 11

1. e

2. 3/2

Practice exercise 12

1. −16

Practice exercise 13

1. 2 + 1 = 3

2. ∫ from 0 to 2 = −10/3; from 2 to 3 = 11/6: 31/6

Practice exercise 14

1. e + 1/e − 2 = 1.09

2. 9/2

Practice exercise 15

1. ∫ from 0 to 8 of |v| dt = 69.2 (technology)

Practice exercise 16

1. 6000(e0.2 − 1) = 1328

Practice exercise 17

1. 0.773

Practice exercise 18

1. k2 + k = 12: k = 3

Practice exercise 19

1. intersections −1, 3: 32/3

Practice exercise 20

1. ln x

DP MATHEMATICS AA · TOPIC 5: CALCULUS

5.16

Further Integration and Volumes of Revolution

Full notes for HL · Syllabus AHL 5.16, 5.17

Contents

1	Further standard integrals
2	Integration by substitution
3	Definite integrals with substitution
4	Integration by parts
5	Repeated integration by parts
6	Partial fractions and trigonometric identities in integration
7	Areas with respect to y
8	Volumes of revolution
9	Substitution: choosing u
10	Definite integrals by substitution: changing limits
11	Integrals giving arctan and arcsin
12	Integration by parts: choosing u
13	Repeated parts and cyclic integrals
14	Partial fractions in integration
15	Trigonometric integrals using identities
16	Volumes of revolution: more practice
17	Areas with respect to y and between curves
18	Mixed review: worked problems
19	Additional exam-style practice
20	Choosing an integration technique
21	Exam-style questions with markscheme
22	Common misconceptions and links
23	Check your understanding
24	Summary and topic map
25	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
integrate sec2x, ax, 1/(a2 + x2) and 1/√(a2 − x2)
use substitution for indefinite and definite integrals, changing the limits
use integration by parts, including repeated use and the 'reappearing integral'
use partial fractions and trigonometric identities to integrate
find areas between a curve and the y-axis
find volumes of revolution about the x-axis and the y-axis

This booklet covers Additional Higher Level (AHL) content (AHL 5.16, 5.17). It is studied by HL students only.

Guiding questions

How can more complicated functions be integrated? How can integration measure the volume of a solid?

How this topic is examined

Further integration is a major HL topic on both papers: substitution (often given), integration by parts, the arctan and arcsin integrals, and volumes of revolution. Paper 1 requires exact answers; Paper 2 uses technology for volumes and areas.

When substituting in a definite integral, change the limits to the new variable, so you do not need to substitute back.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 Further standard integrals

Reversing the further derivatives.

f(x)	∫f(x) dx
sec2x	tan x + c
ax	ax/ln a + c
1/(a2 + x2)	(1/a) arctan(x/a) + c
1/√(a2 − x2)	arcsin(x/a) + c, \|x\| < a

WORKED EXAMPLE 1 arctan and arcsin forms (standard)

Find (a) ∫1/(9 + x2) dx; (b) ∫1/√(16 − x2) dx; (c) ∫ from 0 to 1 of 1/(1 + x2) dx.

Solution

(a) (1/3)arctan(x/3) + cA1

(b) arcsin(x/4) + cA1

WORKED EXAMPLE 2 Completing the square (exam level)

Find ∫1/(x2 + 4x + 13) dx.

Solution

x2 + 4x + 13 = (x + 2)2 + 9M1

= (1/3)arctan((x + 2)/3) + cA1

Practice exercise 1

1. Find ∫1/(4 + x2) dx.

2. Find ∫3/√(25 − x2) dx.

3. Find ∫2x dx.

4. Find ∫1/(1 + 9x2) dx.

2 Integration by substitution

Change the variable.

Choose u = g(x) (often the 'inside' of a composite, or a bracket raised to a power), find du = g′(x) dx, and rewrite the whole integral in terms of u. If the integral cannot be written entirely in u, the substitution is not suitable. In exams, the substitution is usually given.

WORKED EXAMPLE 3 Given substitution (standard)

Use u = 2x + 1 to find ∫x√(2x + 1) dx.

Solution

x = (u − 1)/2, dx = du/2M1

∫((u − 1)/2)u1/2(du/2) = ¼∫(u3/2 − u1/2) duA1

= ¼(2u5/2/5 − 2u3/2/3) = (2x + 1)5/2/10 − (2x + 1)3/2/6 + cA1

WORKED EXAMPLE 4 Trig substitution (exam level)

Use u = sin x to find ∫cos3x dx.

Solution

cos3x = (1 − sin2x)cos x; du = cos x dxM1

∫(1 − u2) du = u − u3/3A1

= sin x − sin3x/3 + cA1

Practice exercise 2

1. Use u = x2 + 1 to find ∫x3/(x2 + 1) dx.

2. Use u = ln x to find ∫(ln x)2/x dx.

3. Use u = ex to find ∫ex/(1 + e2x) dx.

3 Definite integrals with substitution

Change the limits too.

WORKED EXAMPLE 5 Changing limits (standard)

Find ∫ from 0 to 2 of x(x2 + 1)3 dx using u = x2 + 1.

Solution

du = 2x dx; x = 0 → u = 1; x = 2 → u = 5M1 A1

½∫ from 1 to 5 of u3 du = ½[u4/4] from 1 to 5A1

= (625 − 1)/8 = 78A1

Practice exercise 3

1. Evaluate ∫ from 0 to π/2 of sin x cos2x dx using u = cos x.

2. Evaluate ∫ from 1 to e of ln x/x dx.

4 Integration by parts

Reverse the product rule.

Why it works

The product rule gives d/dx(uv) = u(dv/dx) + v(du/dx). Integrating both sides, uv = ∫u(dv/dx) dx + ∫v(du/dx) dx; rearranging gives the formula.

Choosing u: pick u to be the function that becomes simpler when differentiated. A useful priority order is LIATE: Logarithms, Inverse trig, Algebraic (powers of x), Trigonometric, Exponential. For ∫ln x dx, write it as ∫1 × ln x dx with u = ln x.

WORKED EXAMPLE 6 A power of x times an exponential (basic)

Find ∫xe2x dx.

Solution

u = x, dv/dx = e2x: du/dx = 1, v = e2x/2M1

= xe2x/2 − ∫e2x/2 dxA1

= xe2x/2 − e2x/4 + cA1

WORKED EXAMPLE 7 ln x (standard)

Find ∫ln x dx.

Solution

u = ln x, dv/dx = 1: v = xM1

= x ln x − ∫x(1/x) dxA1

= x ln x − x + cA1

WORKED EXAMPLE 8 Definite integral by parts (exam level)

Find ∫ from 0 to π/2 of x cos x dx exactly.

Solution

u = x, v = sin x: [x sin x] − ∫sin x dxM1

= [x sin x + cos x] from 0 to π/2A1

= (π/2 + 0) − (0 + 1) = π/2 − 1A1

Practice exercise 4

1. Find ∫x sin 3x dx.

2. Find ∫x2 ln x dx.

3. Find ∫arctan x dx.

5 Repeated integration by parts

Twice, or until the integral reappears.

For x2ex or x2sin x, apply parts twice, reducing the power of x each time. For exsin x, applying parts twice brings back the original integral I; then solve the equation for I.

WORKED EXAMPLE 9 Reappearing integral (exam level)

Find I = ∫exsin x dx.

Solution

u = sin x, v = ex: I = exsin x − ∫excos x dxM1

Again, u = cos x: ∫excos x dx = excos x + ∫exsin x dx = excos x + IM1 A1

I = exsin x − excos x − I, so I = ½ex(sin x − cos x) + cA1

Practice exercise 5

1. Find ∫x2ex dx.

2. Find ∫e2xcos x dx.

6 Partial fractions and trigonometric identities in integration

Rewrite, then integrate.

Rational functions with factorizable denominators are split into partial fractions (1.11), each integrating to a logarithm. Powers of trigonometric functions are rewritten with identities: sin2x = (1 − cos 2x)/2, cos2x = (1 + cos 2x)/2, tan2x = sec2x − 1.

WORKED EXAMPLE 10 Partial fractions (standard)

Find ∫(5x + 1)/((x − 1)(x + 2)) dx.

Solution

= ∫[2/(x − 1) + 3/(x + 2)] dxM1 A1

= 2 ln|x − 1| + 3 ln|x + 2| + cA1

WORKED EXAMPLE 11 Double angle (standard)

Find ∫ from 0 to π of sin2x dx.

Solution

= ∫(1 − cos 2x)/2 dxM1

= [x/2 − sin 2x/4] from 0 to πA1

= π/2A1

Practice exercise 6

1. Find ∫cos23x dx.

2. Find ∫tan2x dx.

3. Find ∫1/(x2 − 1) dx.

7 Areas with respect to y

Integrate along the y-axis.

The area between a curve and the y-axis, from y = c to y = d, is ∫ from c to d of x dy, where x is written as a function of y. This is useful when the curve is more easily written as x = g(y).

WORKED EXAMPLE 12 Area against the y-axis (standard)

Find the area between y = √x, the y-axis and the line y = 2.

Solution

x = y2M1

∫ from 0 to 2 of y2 dy = [y3/3]A1

= 8/3A1

8 Volumes of revolution

Stacking circular discs.

When the region under y = f(x), from x = a to x = b, is rotated through 2π about the x-axis, each thin slice becomes a disc of radius y and thickness δx, with volume πy2δx. Summing and taking the limit:

For rotation about the y-axis, use x as a function of y. For a region between two curves rotated about the x-axis, subtract volumes: π∫(y12 − y22) dx (washers).

Figure 1. Rotating y = √x about the x-axis forms a solid made of circular discs.

WORKED EXAMPLE 13 About the x-axis (standard)

Find the volume when the region under y = √x, 0 ≤ x ≤ 4, is rotated 2π about the x-axis.

Solution

V = π∫ from 0 to 4 of x dxM1

= π[x2/2] from 0 to 4A1

= 8πA1

WORKED EXAMPLE 14 About the y-axis (exam level)

The region between y = x2, the y-axis and y = 4 (x ≥ 0) is rotated about the y-axis. Find the volume.

Solution

x2 = yM1

V = π∫ from 0 to 4 of y dy = π[y2/2]A1

= 8πA1

WORKED EXAMPLE 15 Between two curves (exam level)

The region between y = x and y = x2 is rotated about the x-axis. Find the volume.

Solution

Intersect at x = 0, 1; y = x is aboveA1

V = π∫ from 0 to 1 of (x2 − x4) dxM1

= π(1/3 − 1/5) = 2π/15A1

Common mistake

Using π∫(y1 − y2)2 dx for a region between curves. The correct washer volume is π∫(y12 − y22) dx.

Practice exercise 7

1. Find the volume when y = ex, 0 ≤ x ≤ 1, is rotated about the x-axis.

2. Show that rotating y = √(r2 − x2) about the x-axis gives a sphere of volume (4/3)πr3.

3. Find the volume when y = sin x, 0 ≤ x ≤ π, is rotated about the x-axis.

9 Substitution: choosing u

What to substitute.

Good choices for u: the inside of a power or root, the exponent of e, the argument of ln, or a function whose derivative appears as a factor. After substituting, the integral must contain only u.

WORKED EXAMPLE 16 Root (exam level)

Find ∫x3√(x2 + 1) dx using u = x2 + 1.

Solution

x2 = u − 1, x dx = du/2M1

½∫(u − 1)u1/2 du = ½(2u5/2/5 − 2u3/2/3)A1

= (x2 + 1)5/2/5 − (x2 + 1)3/2/3 + cA1

Practice exercise 8

1. Find ∫ ex/(1 + ex) dx.

2. Find ∫ x/√(x + 4) dx using u = x + 4.

10 Definite integrals by substitution: changing limits

No need to substitute back.

When substituting in a definite integral, convert the limits to values of u. Then evaluate directly in u.

WORKED EXAMPLE 17 Changing limits (standard)

Evaluate ∫ from 0 to π/2 of cos x esin x dx.

Solution

u = sin x: limits 0 to 1M1

∫ from 0 to 1 of eu du = e − 1A1

Practice exercise 9

1. Evaluate ∫ from 1 to 2 of x(x2 − 1)3 dx.

2. Evaluate ∫ from 0 to 1 of x/(x2 + 1) dx.

11 Integrals giving arctan and arcsin

Recognize the forms.

∫1/(a2 + x2) dx = (1/a) arctan(x/a) + c; ∫1/√(a2 − x2) dx = arcsin(x/a) + c. Complete the square to match these forms.

WORKED EXAMPLE 18 Completing the square (exam level)

Find ∫1/(x2 − 6x + 13) dx.

Solution

(x − 3)2 + 4M1

½ arctan((x − 3)/2) + cA1

WORKED EXAMPLE 19 arcsin form (standard)

Evaluate ∫ from 0 to 1 of 1/√(4 − x2) dx.

Solution

[arcsin(x/2)] from 0 to 1M1

= π/6A1

Practice exercise 10

1. Find ∫ 3/(9 + 4x2) dx.

2. Find ∫ 1/√(5 − 4x − x2) dx.

12 Integration by parts: choosing u

LIATE.

Choose u in the order Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Then dv/dx is the rest. Apply ∫u dv = uv − ∫v du.

WORKED EXAMPLE 20 Inverse trig (exam level)

Find ∫arcsin x dx.

Solution

u = arcsin x, v = xM1

x arcsin x − ∫x/√(1 − x2) dxA1

= x arcsin x + √(1 − x2) + cA1

Practice exercise 11

1. Find ∫ x e−x dx.

2. Find ∫ x2 ln x dx.

3. Find ∫ x sec2x dx.

13 Repeated parts and cyclic integrals

Twice, or solve for I.

For x2ex or x2 sin x, apply parts twice. For eax sin bx, applying parts twice returns the original integral; solve the resulting equation for it.

WORKED EXAMPLE 21 Cyclic (exam level)

Find I = ∫e2x cos x dx.

Solution

I = e2x sin x − 2∫e2x sin x dxM1

∫e2x sin x dx = −e2x cos x + 2IM1

I = e2x sin x + 2e2x cos x − 4I: I = e2x(sin x + 2 cos x)/5 + cA1

Practice exercise 12

1. Find ∫ x2 sin x dx.

2. Find ∫ ex sin x dx.

14 Partial fractions in integration

Split, then integrate.

Split a rational function into partial fractions (1.11); each linear factor gives a logarithm. Divide first if the fraction is improper.

WORKED EXAMPLE 22 Partial fractions (standard)

Find ∫(3x + 1)/((x + 1)(x − 1)) dx.

Solution

= 1/(x + 1) + 2/(x − 1)M1 A1

= ln|x + 1| + 2 ln|x − 1| + cA1

Practice exercise 13

1. Find ∫ 1/(x2 − x) dx.

2. Evaluate ∫ from 2 to 3 of 2/(x2 − 1) dx.

15 Trigonometric integrals using identities

Powers of sin and cos.

sin2x = (1 − cos 2x)/2 and cos2x = (1 + cos 2x)/2 integrate easily. For odd powers, split off one factor and use sin2 + cos2 = 1 with substitution. tan2x = sec2x − 1.

WORKED EXAMPLE 23 Odd power (exam level)

Find ∫sin3x dx.

Solution

∫(1 − cos2x) sin x dx; u = cos xM1

= −cos x + cos3x/3 + cA1

Practice exercise 14

1. Find ∫ cos2(2x) dx.

2. Evaluate ∫ from 0 to π/4 of tan2x dx.

16 Volumes of revolution: more practice

Discs and washers.

About the x-axis: V = π∫y2 dx; about the y-axis: V = π∫x2 dy. For a region between two curves, subtract: V = π∫(y12 − y22) dx.

WORKED EXAMPLE 24 About the y-axis (exam level)

The region bounded by y = ln x, the y-axis, y = 0 and y = 2 is rotated about the y-axis. Find the volume.

Solution

x = ey: V = π∫ from 0 to 2 of e2y dyM1 A1

= π(e4 − 1)/2 ≈ 84.2A1

WORKED EXAMPLE 25 Washer (exam level)

The region between y = √x and y = x/2 is rotated about the x-axis. Find the volume.

Solution

Intersect at x = 0, 4A1

π∫ from 0 to 4 of (x − x2/4) dxM1

= π(8 − 16/3) = 8π/3A1

Practice exercise 15

1. Find the volume when y = sec x, 0 ≤ x ≤ π/4, is rotated about the x-axis.

2. Find the volume of a cone of radius r and height h by rotating y = rx/h.

17 Areas with respect to y and between curves

Choosing the variable.

When a region is bounded by curves easier to write as x = g(y), integrate with respect to y: area = ∫(right − left) dy.

WORKED EXAMPLE 26 With respect to y (standard)

Find the area between x = y2 and x = 2y.

Solution

Intersect at y = 0, 2A1

∫ from 0 to 2 of (2y − y2) dy = 4/3M1 A1

Practice exercise 16

1. Find the area between y = √x, the y-axis and y = 3.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 27 Mixed problem (exam level)

(a) Find ∫x ex dx. (b) The region under y = √(x) ex/2 for 0 ≤ x ≤ 1 is rotated about the x-axis. Find the volume exactly.

Solution

(a) ex(x − 1) + cM1 A1

(b) π∫ from 0 to 1 of xex dx = π[ex(x − 1)] from 0 to 1M1

= πA1

Practice exercise 17

1. Find ∫ ln x/x2 dx.

19 Additional exam-style practice

Paper 1.

WORKED EXAMPLE 28 Paper 1 question (exam level)

(a) Express (x + 7)/((x − 1)(x + 2)) in partial fractions. (b) Hence find ∫ from 2 to 4 of (x + 7)/((x − 1)(x + 2)) dx exactly.

Solution

(a) 8/(3(x − 1)) − 5/(3(x + 2))M1 A1

(b) (8/3) ln 3 − (5/3) ln(6/4)M1

= (8/3) ln 3 − (5/3) ln(3/2)A1

Practice exercise 18

1. Find ∫ from 0 to 1 of 1/(1 + x2) dx and hence an estimate of π using technology.

20 Choosing an integration technique

A decision guide.

Integrand looks like	Try
f′(x) × g(f(x))	substitution / inspection
product of different types (x ex, x ln x)	by parts
rational with factorizable denominator	partial fractions
1/(a2 + x2), 1/√(a2 − x2)	arctan / arcsin
sin2, cos2, tan2	identities

WORKED EXAMPLE 29 Choosing (standard)

Choose a method for ∫x cos(x2) dx and ∫x cos x dx.

Solution

First: substitution u = x2; second: by partsR1 R1

21 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 15 minutes)

(a) Use integration by parts to find ∫x ln x dx. [4]

(b) Hence find the exact value of ∫ from 1 to e of x ln x dx. [2]

Markscheme

Part	Answer	Marks
(a)	u = ln x, v = x2/2	M1
	(x2/2)ln x − ∫x/2 dx	A1
	(x2/2)ln x − x2/4 + c	A1 A1
(b)	(e2/2 − e2/4) − (0 − 1/4)	M1
	(e2 + 1)/4	A1
(c)	du = 2x dx; x2 = u − 1; limits 1 to 2	M1 A1
	½∫ from 1 to 2 of (u − 1)/u du	A1
	½[u − ln u] from 1 to 2	M1
	½(1 − ln 2)	A1

Examiner's comments. Changing the limits in (c) avoids substituting back.

Question 2 (Paper 2 style) (about 10 minutes)

The region R is bounded by y = 2 − x2 and y = x2.

(a) Find the area of R. [3]

(b) Find the volume when R is rotated through 2π about the x-axis. [3]

Markscheme

Part	Answer	Marks
(a)	intersections x = ±1	A1
	∫ from −1 to 1 of (2 − 2x2) dx	M1
	8/3	A1
(b)	π∫ from −1 to 1 of ((2 − x2)2 − x4) dx	M1 A1
	16π/3 ≈ 16.8	A1

Examiner's comments. The washer method subtracts the squares of the two functions.

22 Common misconceptions and links

Students often think…	Actually…
∫1/(a2 + x2) dx = ln(a2 + x2)	(1/a)arctan(x/a)
forget to change limits	change them or substitute back
choose u = ex in parts	LIATE: choose the algebraic or log factor
washer volume uses (y1 − y2)2	use y12 − y22

How 5.16 connects to the rest of the course

Topic	Connection
5.5 Integration	basic integrals
1.11 Partial fractions	splitting rationals
5.14 Further derivatives	arctan and arcsin
5.18 Differential equations	separating variables

23 Check your understanding

Section A: multiple choice

1. ∫1/(1 + x2) dx =

A. arctan x + c B. ln(1 + x2) + c C. arcsin x + c D. 1/x + c

2. Integration by parts reverses

A. the chain rule B. the quotient rule C. the product rule D. the power rule

3. ∫ln x dx =

A. 1/x + c B. x ln x − x + c C. x ln x + c D. (ln x)2/2 + c

4. Volume about the x-axis:

A. ∫πy dx B. 2π∫y dx C. π∫x2 dx D. π∫y2 dx

5. ∫sec2x dx =

A. sec x + c B. tan x + c C. −cot x + c D. sec3x/3 + c

Section B: structured

6. Find ∫x2cos x dx. [4]

7. Find the volume when y = 1/x, 1 ≤ x ≤ 3, is rotated about the x-axis. [3]

Answers

1. A 2. C 3. B 4. D 5. B

6. x2sin x + 2x cos x − 2 sin x + c.

7. π[−1/x] from 1 to 3 = 2π/3.

24 Summary and topic map

Idea	Remember
Further integrals	sec2x; ax; arctan; arcsin forms
Substitution	u = g(x); change limits
By parts	∫udv = uv − ∫vdu; LIATE
Repeated parts	reduce powers or solve for the reappearing integral
Identities	sin2, cos2 via cos 2x
Area with y	∫x dy
Volumes	π∫y2 dx; π∫x2 dy; washers

Before the exam, make sure you…

•choose u carefully
•change limits in substitution
•complete the square for arctan forms
•square before subtracting in washers

Further Integration and Volumes on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 2. Topic map for Further Integration and Volumes.

25 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. ½ arctan(x/2) + c

2. 3 arcsin(x/5) + c

3. 2x/ln 2 + c

4. (1/3)arctan 3x + c

Practice exercise 2

1. ½(x2 + 1) − ½ ln(x2 + 1) + c

2. (ln x)3/3 + c

3. arctan(ex) + c

Practice exercise 3

1. 1/3

2. 1/2

Practice exercise 4

1. −(x/3)cos 3x + (1/9)sin 3x + c

2. (x3/3)ln x − x3/9 + c

3. x arctan x − ½ ln(1 + x2) + c

Practice exercise 5

1. ex(x2 − 2x + 2) + c

2. e2x(2 cos x + sin x)/5 + c

Practice exercise 6

1. x/2 + sin 6x/12 + c

2. tan x − x + c

3. ½ ln|(x − 1)/(x + 1)| + c

Practice exercise 7

1. π(e2 − 1)/2

2. π∫ from −r to r of (r2 − x2) dx = (4/3)πr3

3. π2/2

Practice exercise 8

1. ln(1 + ex) + c

2. (2/3)(x + 4)3/2 − 8(x + 4)1/2 + c

Practice exercise 9

1. 81/8

2. ½ ln 2

Practice exercise 10

1. ½ arctan(2x/3) + c

2. arcsin((x + 2)/3) + c

Practice exercise 11

1. −e−x(x + 1) + c

2. x3 ln x/3 − x3/9 + c

3. x tan x + ln|cos x| + c

Practice exercise 12

1. −x2 cos x + 2x sin x + 2 cos x + c

2. ex(sin x − cos x)/2 + c

Practice exercise 13

1. ln|(x − 1)/x| + c

2. ln(3/2)

Practice exercise 14

1. x/2 + sin 4x/8 + c

2. 1 − π/4

Practice exercise 15

1. π

2. πr2h/3

Practice exercise 16

1. ∫ from 0 to 3 of y2 dy = 9

Practice exercise 17

1. −(ln x + 1)/x + c

Practice exercise 18

1. π/4

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