Integration by partsAQA A-Level Maths: Subtopic test

1. 1
   Let $I=\int x\,e^{2x}\,dx$.

   (a)

   Which choice of $u$ and $\frac{dv}{dx}$ is best for integration by parts?

   [1 mark]

   • A$u=e^{2x},\ \frac{dv}{dx}=x$
   • B$u=xe^{2x},\ \frac{dv}{dx}=1$
   • C$u=2x,\ \frac{dv}{dx}=e^{x}$
   • D$u=x,\ \frac{dv}{dx}=e^{2x}$

   (b)

   Find $I$.

   [1 mark]

   • A$\frac{e^{2x}}{4}(2x-1)+c$
   • B$\frac{e^{2x}}{2}(x-1)+c$
   • C$\frac{e^{2x}}{2}(x+1)+c$
   • D$xe^{2x}-\frac{e^{2x}}{2}+c$

   (c)

   Hence find the exact value of $\int_0^1x\,e^{2x}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student tries to find $\int x\cos x\,dx$ by integration by parts, choosing $u=\cos x$ and $\frac{dv}{dx}=x$.

   (a)

   What is wrong with the student's choice?

   [1 mark]

   • AIntegration by parts cannot be used when both functions are of $x$.
   • BIt leads to $\int\frac{x^2}{2}\sin x\,dx$, which is harder than the original integral.
   • CNothing: this choice always gives a simpler integral than $u=x$.
   • D$\cos x$ cannot be differentiated.

   (b)

   Find $\int x\cos x\,dx$.

   [1 mark]

   • A$x\sin x-\cos x+c$
   • B$\frac{x^2}{2}\sin x+c$
   • C$x\sin x+\cos x+c$
   • D$-x\sin x+\cos x+c$

   (c)

   Hence find the exact value of $\int_0^{\pi/2}x\cos x\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $y=\ln x$ is defined for $x>0$.

   (a)

   Use integration by parts with $u=\ln x$ to show that $\int\ln x\,dx=x\ln x-x+c$.

   [3 marks]

   (b)

   Hence find the exact area of the region under the curve $y=\ln x$ between $x=1$ and $x=e^2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Give exact answers in each part.

   (a)

   Use integration by parts twice to find $\int_0^1x^2e^x\,dx$.

   [6 marks]

   (b)

   Find $\int_1^ex^2\ln x\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions