Integration by substitutionAQA A-Level Maths: Subtopic test

1. 1
   Let $I=\int 6x\,(x^2+5)^3\,dx$.

   (a)

   Which substitution is most suitable for finding $I$?

   [1 mark]

   • A$u=x$
   • B$u=6x$
   • C$u=x^2+5$
   • D$u=(x^2+5)^3$

   (b)

   Find $I$.

   [1 mark]

   • A$\frac14(x^2+5)^4+c$
   • B$3(x^2+5)^4+c$
   • C$\frac{6x(x^2+5)^4}{4}+c$
   • D$\frac34(x^2+5)^4+c$

   (c)

   Hence find the exact value of $\int_0^16x\,(x^2+5)^3\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $J=\int_0^{\pi/2}\sin x\cos^3x\,dx$, which is to be found using the substitution $u=\cos x$.

   (a)

   Which integral is equal to $J$ after the substitution?

   [1 mark]

   • A$\int_0^1u^3\,du$
   • B$\int_0^{\pi/2}u^3\,du$
   • C$\int_1^0u^3\,du$
   • D$\int_0^1(-u^3)\,du$

   (b)

   Find the value of $J$.

   [1 mark]

   • A$-\frac14$
   • B$\frac14$
   • C$\frac13$
   • D$\frac12$

   (c)

   Explain why $J$ is positive even though the substitution gives $du=-\sin x\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $K=\int\frac{x}{\sqrt{x+4}}\,dx$, for $x>-4$, which is to be found using the substitution $u=x+4$.

   (a)

   Use the substitution to show that $K=\frac23(x+4)^{\frac32}-8(x+4)^{\frac12}+c$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^5\frac{x}{\sqrt{x+4}}\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In each part a suitable substitution is given. Give exact answers.

   (a)

   Use the substitution $u=1+e^x$ to show that $\int_0^{\ln2}\frac{e^{2x}}{1+e^x}\,dx=1+\ln\frac23$.

   [6 marks]

   (b)

   Use the substitution $u=\cos x$ to show that $\int_0^{\pi/3}\tan x\,dx=\ln2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions