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Integration and Its Applications · Paper 3 — Free Maths AA HL Practice Questions

1ChallengeLAQIndefinite integrals and their propertiess10 marksPaper 3~15 min
Consider the function ff defined by f(x)=2x3−5x2+5x−3x2−2x+1f(x) = \dfrac{2x^3 - 5x^2 + 5x - 3}{x^2 - 2x + 1}, for x≠1x \neq 1.
(a)
Prove that ∫f(x) dx=x2−x+ln⁡∣x−1∣+1x−1+C\displaystyle\int f(x)\,dx = x^2 - x + \ln|x-1| + \frac{1}{x-1} + C, where CC is an arbitrary constant. [6 marks]
(b)
Hence, evaluate ∫24f(x) dx\displaystyle\int_2^4 f(x)\,dx, justifying why the Fundamental Theorem of Calculus applies. [4 marks]
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