Integration by parts

Integration

A-level Maths (9MA0)

ORIGINAL

Choose u to be the part that simplifies when you differentiate it — then it's just the formula.

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Question

Find

\displaystyle\int x e^x \, dx

Every step worked, with the reasoning.

1
Let $u = x$ and $\dfrac{dv}{dx} = e^x$ .
Pick $u = x$ because differentiating it simplifies to 1 (LATE/LIATE).
2
$\dfrac{du}{dx} = 1$ and $v = e^x$
Differentiate $u$ ; integrate $\frac{dv}{dx}$ .
3
$\int x e^x\,dx = uv - \int v\,\dfrac{du}{dx}\,dx = x e^x - \int e^x\,dx$
Apply $\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$ .
4
$= x e^x - e^x + C = e^x(x - 1) + C$
Integrate $e^x$ and add the constant.

Answer: $e^x(x - 1) + C$

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