Integration techniques
Integration
- A2
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
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Find $\displaystyle \int 2x(x^2 + 1)^3\,dx$.
3 marks
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Find $\displaystyle \int xe^x\,dx$.
3 marks
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Find $\displaystyle \int \dfrac{1}{(x - 1)(x + 2)}\,dx$.
4 marks
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Evaluate $\displaystyle \int_1^e \ln x\,dx$.
4 marks
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Find $\displaystyle \int \dfrac{x}{\sqrt{x^2 + 1}}\,dx$.
5 marks
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Find $\displaystyle \int xe^{2x}\,dx$.
5 marks
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Find $\displaystyle \int \dfrac{x + 7}{(x - 1)(x + 3)}\,dx$.
6 marks
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Evaluate $\displaystyle \int_0^1 \dfrac{x + 3}{(x + 1)(x + 2)}\,dx$.
7 marks
Mark scheme
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Answer. $\dfrac{1}{4}(x^2 + 1)^4 + c$
- Use the substitution $u = x^2 + 1$, so $du = 2x\,dx$.
- $\displaystyle \int u^3\,du = \dfrac{u^4}{4}$.
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Answer. $e^x(x - 1) + c$
- Use integration by parts with $u = x$ and $\dfrac{dv}{dx} = e^x$.
- $xe^x - \displaystyle \int e^x\,dx$.
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Answer. $\dfrac{1}{3}\ln\left|\dfrac{x - 1}{x + 2}\right| + c$
- Split as $\dfrac{1/3}{x - 1} - \dfrac{1/3}{x + 2}$.
- Integrate to a difference of logarithms.
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Answer. $1$
- Integration by parts gives $x\ln x - x$.
- The limits give $(e - e) - (0 - 1) = 1$.
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Answer. $\sqrt{x^2 + 1} + c$
- Use $u = x^2 + 1$, so $du = 2x\,dx$.
- $\dfrac{1}{2}\displaystyle \int u^{-1/2}\,du = \sqrt{u}$.
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Answer. $\dfrac{e^{2x}(2x - 1)}{4} + c$
- Parts with $u = x$ gives $\dfrac{x}{2}e^{2x} - \dfrac{1}{2}\displaystyle \int e^{2x}\,dx$.
- This simplifies to $\dfrac{e^{2x}(2x - 1)}{4} + c$.
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Answer. $2\ln|x - 1| - \ln|x + 3| + c$
- Split as $\dfrac{2}{x - 1} - \dfrac{1}{x + 3}$.
- Integrate each term.
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Answer. $\ln \dfrac{8}{3}$
- Split as $\dfrac{2}{x + 1} - \dfrac{1}{x + 2}$.
- The antiderivative is $2\ln|x + 1| - \ln|x + 2|$.
- The limits give $(2\ln 2 - \ln 3) - (-\ln 2) = \ln \dfrac{8}{3}$.