What is the integral from 0 to pi of xsin(4x) ?

The integral from 0 to pi of xsin(4x) is -pi/4

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integral from 0 to pi of xsin(4x)

\int_{0}^{π} x sin (4 x) d x

- \frac{π}{4}

Decimal Notation

- 0.78539 \dots

Solution steps

\int_{0}^{π} x sin (4 x) d x

Apply Integration By Parts

: [\frac{1}{4} (- x cos (4 x) - 4 \cdot \int - \frac{1}{4} cos (4 x) d x)]_{0}^{π}

= [\frac{1}{4} (- x cos (4 x) - 4 \cdot \int - \frac{1}{4} cos (4 x) d x)]_{0}^{π}

\int - \frac{1}{4} cos (4 x) d x = - \frac{1}{16} sin (4 x)

= [\frac{1}{4} (- x cos (4 x) - 4 (- \frac{1}{16} sin (4 x)))]_{0}^{π}

Simplify

[\frac{1}{4} (- x cos (4 x) - 4 (- \frac{1}{16} sin (4 x)))]_{0}^{π} : [\frac{1}{4} (- x cos (4 x) + \frac{1}{4} sin (4 x))]_{0}^{π}

= [\frac{1}{4} (- x cos (4 x) + \frac{1}{4} sin (4 x))]_{0}^{π}

Compute the boundaries:

- \frac{π}{4}

= - \frac{π}{4}

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What is the integral from 0 to pi of xsin(4x) ?
The integral from 0 to pi of xsin(4x) is -pi/4