Integration by substitutionAQA A-Level Maths: Mind map

Exam questions on Integration by substitution

1. Let $I=\int 6x\,(x^2+5)^3\,dx$.
   Hence find the exact value of $\int_0^16x\,(x^2+5)^3\,dx$.2 marks
2. Let $J=\int_0^{\pi/2}\sin x\cos^3x\,dx$, which is to be found using the substitution $u=\cos x$.
   Explain why $J$ is positive even though the substitution gives $du=-\sin x\,dx$.2 marks
3. Let $K=\int\frac{x}{\sqrt{x+4}}\,dx$, for $x>-4$, which is to be found using the substitution $u=x+4$.
   Use the substitution to show that $K=\frac23(x+4)^{\frac32}-8(x+4)^{\frac12}+c$.3 marks