Integration by partsEdexcel International A Level Maths: Subtopic test

1. 1
   Let $I=\int xe^{3x}\,dx$.

   (a)

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

   [1 mark]

   • A$u=x,\ \frac{dv}{dx}=e^{3x}$
   • B$u=e^{3x},\ \frac{dv}{dx}=x$
   • C$u=xe^{3x},\ \frac{dv}{dx}=1$
   • D$u=3x,\ \frac{dv}{dx}=e^{x}$

   (b)

   Find $I$.

   [1 mark]

   • A$\frac13xe^{3x}-\frac13e^{3x}+c$
   • B$\frac13xe^{3x}+\frac19e^{3x}+c$
   • C$xe^{3x}-\frac13e^{3x}+c$
   • D$\frac13xe^{3x}-\frac19e^{3x}+c$

   (c)

   Hence find the exact value of $\int_0^1xe^{3x}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I=\int\ln x\,dx$, for $x>0$.

   (a)

   Writing $\ln x$ as $1\times\ln x$ with $u=\ln x$ and $\frac{dv}{dx}=1$, which is the first line of the integration by parts?

   [1 mark]

   • A$x\ln x+\int1\,dx$
   • B$\frac{x^2}{2}\ln x-\int\frac{x}{2}\,dx$
   • C$x\ln x-\int1\,dx$
   • D$\ln x-\int\frac1x\,dx$

   (b)

   Hence find $I$.

   [1 mark]

   • A$x\ln x+x+c$
   • B$x\ln x-x+c$
   • C$x\ln x-\frac{x^2}{2}+c$
   • D$\frac1x+c$

   (c)

   The region $R$ is bounded by the curve $y=\ln x$, the $x$-axis and the line $x=e$. Find the area of $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^1x^2e^{x}\,dx$.

   (a)

   Show that $I=e-2\int_0^1xe^{x}\,dx$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=e^{x}\sin x$ for $0\le x\le2\pi$.

   (a)

   Use integration by parts twice to show that $\int e^{x}\sin x\,dx=\frac12e^{x}(\sin x-\cos x)+c$.

   [6 marks]

   (b)

   Find the exact total area of the finite regions bounded by $C$ and the $x$-axis for $0\le x\le2\pi$.

   [6 marks]

   Total for question 4: 12 marks

End of questions