Math · Calculus II · Worked example

Integrate xeˣ by parts

Find the integral of xeˣ with respect to x.

\int x e^{x} d x

Choose u and dv

x gets simpler when differentiated, and eˣ is easy to integrate.

The parts for the integral of xeˣ
Part	Choice	Then
u	x	du = dx
dv	eˣ dx	v = eˣ

Apply the formula

uv − ∫v du, with the new integral simpler than the old.

\int x e^{x} d x = x e^{x} - \int e^{x} d x

Finish the integral

The integral of eˣ is eˣ.

\int x e^{x} d x = x e^{x} - e^{x} + C

Check by differentiating

The product rule gives eˣ + xeˣ, and subtracting the derivative of eˣ leaves xeˣ.

\frac{d}{d x} (x e^{x} - e^{x}) = x e^{x}

Result

xeˣ − eˣ + C.

Your turn

Find the integral of x cos x.

Show the answer and explanation

x sin x + cos x + C.

u = x and dv = cos x dx give du = dx and v = sin x, so the integral is x sin x minus the integral of sin x, which is x sin x + cos x + C.

\begin{array}{l} \int x cos x d x \\ x sin x + cos x + C \end{array}

Keep exploring

In Derivative & antiderivative checks, enter xeˣ + eˣ instead, a common sign slip. The checker reports that it does not agree: its derivative is xeˣ + 2eˣ.

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Sources and scope

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OpenStax Calculus Volume 2 — Integration by parts

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