Solve \int x cos(2x) dx

Evaluate

\int x cos (2 x) d x

Prepare for integration by parts

u = x d v = cos (2 x) d x

Calculate the derivative

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d u = d x d v = cos (2 x) d x

Evaluate the integral

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d u = d x v = \frac{1}{2} sin (2 x)

Substitute u = x 、 v = \frac{1}{2} sin (2 x) 、 d u = d x 、 d v = cos (2 x) d x for \int u d v = uv - \int v d u

x \times \frac{1}{2} sin (2 x) - \int 1 \times \frac{1}{2} sin (2 x) d x

Calculate

\frac{1}{2} x sin (2 x) - \int \frac{1}{2} sin (2 x) d x

Evaluate the integral

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\frac{1}{2} x sin (2 x) + \frac{1}{4} cos (2 x)

Solution

\frac{1}{2} x sin (2 x) + \frac{1}{4} cos (2 x) + C, C \in R