What is the integral from 0 to pi of x*sin(3x) ?

The integral from 0 to pi of x*sin(3x) is pi/3

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integral from 0 to pi of x*sin(3x)

\int_{0}^{π} x \cdot sin (3 x) d x

\frac{π}{3}

Decimal

1.04719 \dots

Solution steps

\int_{0}^{π} x sin (3 x) d x

Apply Integration By Parts

: [\frac{1}{3} (- x cos (3 x) - 3 \cdot \int - \frac{1}{3} cos (3 x) d x)]_{0}^{π}

= [\frac{1}{3} (- x cos (3 x) - 3 \cdot \int - \frac{1}{3} cos (3 x) d x)]_{0}^{π}

\int - \frac{1}{3} cos (3 x) d x = - \frac{1}{9} sin (3 x)

= [\frac{1}{3} (- x cos (3 x) - 3 (- \frac{1}{9} sin (3 x)))]_{0}^{π}

Simplify

[\frac{1}{3} (- x cos (3 x) - 3 (- \frac{1}{9} sin (3 x)))]_{0}^{π} : [\frac{1}{3} (- x cos (3 x) + \frac{1}{3} sin (3 x))]_{0}^{π}

= [\frac{1}{3} (- x cos (3 x) + \frac{1}{3} sin (3 x))]_{0}^{π}

Compute the boundaries:

\frac{π}{3}

= \frac{π}{3}

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What is the integral from 0 to pi of x*sin(3x) ?
The integral from 0 to pi of x*sin(3x) is pi/3