Integration by partsEdexcel A-Level Maths: Subtopic test

1. 1
   Let $I=\int xe^{2x}\,\mathrm{d}x$, to be found by integration by parts.

   (a)

   Which choice of $u$ and $\frac{\mathrm{d}v}{\mathrm{d}x}$ leads to a simpler integral to evaluate next?

   [1 mark]

   • A$u=e^{2x},\ \frac{\mathrm{d}v}{\mathrm{d}x}=x$
   • B$u=xe^{2x},\ \frac{\mathrm{d}v}{\mathrm{d}x}=1$
   • C$u=1,\ \frac{\mathrm{d}v}{\mathrm{d}x}=xe^{2x}$
   • D$u=x,\ \frac{\mathrm{d}v}{\mathrm{d}x}=e^{2x}$

   (b)

   Find $I$.

   [1 mark]

   • A$\frac14e^{2x}(2x-1)+c$
   • B$\frac12xe^{2x}+\frac14e^{2x}+c$
   • C$\frac12xe^{2x}-\frac12e^{2x}+c$
   • D$xe^{2x}-e^{2x}+c$

   (c)

   Hence find the exact value of $\int_0^1xe^{2x}\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Integration by parts uses $\int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x=uv-\int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x$.

   (a)

   By writing $\ln x=1\times\ln x$ with $u=\ln x$ and $\frac{\mathrm{d}v}{\mathrm{d}x}=1$, find $\int\ln x\,\mathrm{d}x$ for $x>0$.

   [1 mark]

   • A$\frac1x+c$
   • B$x\ln x-x+c$
   • C$x\ln x+c$
   • D$x\ln x-\frac1x+c$

   (b)

   Evaluate $\int_1^e\ln x\,\mathrm{d}x$.

   [1 mark]

   • A$e-1$
   • B$0$
   • C$1$
   • D$e$

   (c)

   Find $\int x^2\ln x\,\mathrm{d}x$ for $x>0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question, use integration by parts.

   (a)

   Find $\int x\sin2x\,\mathrm{d}x$.

   [3 marks]

   (b)

   Find the exact value of $\int_1^3\frac{\ln x}{x^2}\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question, use integration by parts, applying the method more than once where necessary.

   (a)

   Find $\int x^2e^{3x}\,\mathrm{d}x$.

   [6 marks]

   (b)

   Show that $\int_0^{\pi/2}x^2\cos x\,\mathrm{d}x=\frac{\pi^2}{4}-2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions