What do you do with the limits in a definite integral by substitution?

Change them to u-values: x=a becomes u=g(a).

Why expand brackets such as (u+2)\sqrt u before integrating?

Each term becomes a simple power of u that can be integrated.

How can you check an indefinite integral?

Differentiate the answer and compare with the integrand.

Integration by substitutionEdexcel International A Level Maths: Flashcards

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What does integration by substitution reverse?

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What does integration by substitution reverse?: The chain rule.
If $u=g(x)$ , what replaces $dx$ ?: $dx=\frac{du}{g'(x)}$
$\int 2x(x^2+3)^4\,dx$ : which substitution?: $u=x^2+3$ , giving $\int u^4\,du$
Result of $\int 2x(x^2+3)^4\,dx$ ?: $\frac15(x^2+3)^5+c$
$\int x\sqrt{x-2}\,dx$ : what is $x$ in terms of $u=x-2$ ?: $x=u+2$
$\int x\sqrt{x-2}\,dx$ becomes what in terms of $u$ ?: $\int(u+2)\sqrt u\,du$
What do you do with the limits in a definite integral by substitution?: Change them to $u$ -values: $x=a$ becomes $u=g(a)$ .
If $u=3x-1$ , what is $dx$ ?: $dx=\frac13du$
Evaluate $\int_3^4u^4\,du$ .: $\frac{781}{5}$
If $u^2=x+1$ , find $dx$ .: $dx=2u\,du$
Why expand brackets such as $(u+2)\sqrt u$ before integrating?: Each term becomes a simple power of $u$ that can be integrated.
How can you check an indefinite integral?: Differentiate the answer and compare with the integrand.

Exam questions on Integration by substitution

Let $I=\int 2x(x^2+3)^4\,dx$ . Use the substitution $u=x^2+3$ .
Hence find the exact value of $\int_0^1 2x(x^2+3)^4\,dx$ .2 marks
Let $I=\int x\sqrt{x-2}\,dx$ . Use the substitution $u=x-2$ .
Hence find the exact value of $\int_2^6x\sqrt{x-2}\,dx$ .2 marks
Let $I=\int_0^4\frac{x}{\sqrt{2x+1}}\,dx$ .
Using the substitution $u=2x+1$ , show that $I=\frac14\int_1^9\left(u^{1/2}-u^{-1/2}\right)du$ .3 marks

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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).