Integration by substitutionEdexcel A-Level Maths: Subtopic test

1. 1
   Let $I=\int x\left(x^2+3\right)^4\,\mathrm{d}x$.

   (a)

   Which substitution turns $I$ into an integral involving only a power of $u$?

   [1 mark]

   • A$u=x$
   • B$u=x^2+3$
   • C$u=x^4$
   • D$u=2x$

   (b)

   Find $I$.

   [1 mark]

   • A$\frac{\left(x^2+3\right)^5}{5}+c$
   • B$\frac{x^2\left(x^2+3\right)^5}{10}+c$
   • C$\frac{\left(x^2+3\right)^5}{10}+c$
   • D$\left(x^2+3\right)^5+c$

   (c)

   Hence find the exact value of $\int_0^1x\left(x^2+3\right)^4\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $N=\int_0^{\ln2}\frac{e^{x}}{1+e^{x}}\,\mathrm{d}x$, to be found using the substitution $u=1+e^x$.

   (a)

   Find the new limits for $u$.

   [1 mark]

   • A$0$ and $\ln2$
   • B$1$ and $2$
   • C$2$ and $3$
   • D$1$ and $3$

   (b)

   Which integral is equal to $N$?

   [1 mark]

   • A$\int_2^3\frac1u\,\mathrm{d}u$
   • B$\int_0^{\ln2}\frac1u\,\mathrm{d}u$
   • C$\int_2^3\frac{e^x}{u}\,\mathrm{d}u$
   • D$\int_1^2\frac1u\,\mathrm{d}u$

   (c)

   Hence find the exact value of $N$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question, use integration by substitution with the substitution given in each part.

   (a)

   Use $u=1-x^2$ to find $\int\frac{x}{\sqrt{1-x^2}}\,\mathrm{d}x$ for $|x|<1$.

   [3 marks]

   (b)

   Use $u=2x+1$ to find the exact value of $\int_0^4\frac{x}{\sqrt{2x+1}}\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question, use integration by substitution, with the substitution given in each part.

   (a)

   Use $u=\sqrt x$ to show that $\int_1^4\frac{1}{x+\sqrt x}\,\mathrm{d}x=2\ln\frac32$.

   [6 marks]

   (b)

   Use $u=\cos x$ to find the exact value of $\int_0^{\pi/2}\sin^3x\,\mathrm{d}x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions