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

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

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

\int_{0}^{π} t \cdot sin (3 t) d t

\frac{π}{3}

Decimal

1.04719 \dots

Solution steps

\int_{0}^{π} t sin (3 t) d t

Apply Integration By Parts

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

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

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

Simplify

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

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

Compute the boundaries:

\frac{π}{3}

= \frac{π}{3}

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