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Copilot · Practice · IB Maths AA HL

Integration by parts · 4 marks

Find $\int x e^{2 x} d x$

Your working

$u = x, d v = e^{2 x} d x$
$= \frac{1}{2} x e^{2 x} - \int \frac{1}{2} e^{2 x} d x$
$= \frac{1}{2} x e^{2 x} - \frac{1}{2} e^{2 x} + C$

Nearly there: method correct, last step slipped

Integrating $\frac{1}{2} e^{2 x}$ divides by 2 again, so the answer is $\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + C$ .

3 of 4 marksView worked solution

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Worked solution

1
Choose the parts
$u = x, d v = e^{2 x} d x$
2
Differentiate and integrate
$d u = d x, v = \frac{1}{2} e^{2 x}$
3
Apply the formula
$\frac{1}{2} x e^{2 x} - \int \frac{1}{2} e^{2 x} d x$
4
Finish the integral
$\frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + C$

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Study guide · Integration by parts

Key idea

Use it when you're integrating a product. Pick $u$ as the part that gets simpler when you differentiate it.

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

Key ideasFormulaeWorked example

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Compass AI

Why is it

\frac{1}{4} e^{2 x}

? I already halved it once.

You halved it to get

v

. Now you're integrating

\frac{1}{2} e^{2 x}

, a second integral. What happens to the coefficient this time?

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Your progress

Differentiation84%
Integration by partsNeeds practice
Binomial theorem66%
Sequences and series72%

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