How should you choose u?

So that differentiating it makes it simpler (e.g. x or \ln x).

How many times is parts applied to \int x^2e^{x}\,dx?

Twice, since the power of x falls by one each time.

How do you solve a cyclic integral such as \int e^{x}\sin x\,dx?

Apply parts twice, get the original integral on the right and rearrange.

Integration by partsEdexcel International A Level Maths: Flashcards

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State the formula for integration by parts.

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State the formula for integration by parts.: $\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$
Which differentiation rule does integration by parts reverse?: The product rule.
How should you choose $u$ ?: So that differentiating it makes it simpler (e.g. $x$ or $\ln x$ ).
$\int\ln x\,dx$ ?: $x\ln x-x+c$
What choices of $u$ and $\frac{dv}{dx}$ integrate $\ln x$ ?: $u=\ln x$ and $\frac{dv}{dx}=1$
$\int xe^{x}\,dx$ ?: $xe^{x}-e^{x}+c$
$\int xe^{3x}\,dx$ ?: $\frac13xe^{3x}-\frac19e^{3x}+c$
$\int x\cos x\,dx$ ?: $x\sin x+\cos x+c$
How many times is parts applied to $\int x^2e^{x}\,dx$ ?: Twice, since the power of $x$ falls by one each time.
$\int_0^1x^2e^{x}\,dx$ ?: $e-2$
$\int e^{x}\sin x\,dx$ ?: $\frac12e^{x}(\sin x-\cos x)+c$
How do you solve a cyclic integral such as $\int e^{x}\sin x\,dx$ ?: Apply parts twice, get the original integral on the right and rearrange.

Exam questions on Integration by parts

Let $I=\int xe^{3x}\,dx$ .
Hence find the exact value of $\int_0^1xe^{3x}\,dx$ .2 marks
Let $I=\int\ln x\,dx$ , for $x>0$ .
The region $R$ is bounded by the curve $y=\ln x$ , the $x$ -axis and the line $x=e$ . Find the area of $R$ .2 marks
Let $I=\int_0^1x^2e^{x}\,dx$ .
Show that $I=e-2\int_0^1xe^{x}\,dx$ .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).