Integration by substitutionAQA A-Level Maths: Flashcards

What these 12 flashcards ask

What does substitution reverse?
u=x^2+5: what is du?
\int6x(x^2+5)^3dx?
What do you do with the limits of a definite integral after u=g(x)?
\int\frac{f'(x)}{f(x)}dx?
\int\tan x\,dx?
u=\cos x: what is du?
u=x+4: how do you write x?
u=1+e^x: write e^x and du.
What happens if the limits 1 and 0 are reversed?
What must you do with leftover x after substituting?
After an indefinite integral in u, what must you do?

Exam questions on Integration by substitution

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).