What is the integral from 0 to 4 of xsin(3x) ?

The integral from 0 to 4 of xsin(3x) is 1/9 sin(12)-4/3 cos(12)

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integral from 0 to 4 of xsin(3x)

\int_{0}^{4} x sin (3 x) d x

\frac{1}{9} sin (12) - \frac{4}{3} cos (12)

Decimal

- 1.18475 \dots

Solution steps

\int_{0}^{4} x sin (3 x) d x

Apply Integration By Parts

= [- \frac{1}{3} x cos (3 x) - \int - \frac{1}{3} cos (3 x) d x]_{0}^{4}

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

= [- \frac{1}{3} x cos (3 x) - (- \frac{1}{9} sin (3 x))]_{0}^{4}

Simplify

= [- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)]_{0}^{4}

Compute the boundaries:

\frac{1}{9} sin (12) - \frac{4}{3} cos (12)

= \frac{1}{9} sin (12) - \frac{4}{3} cos (12)

What is the integral from 0 to 4 of xsin(3x) ?
The integral from 0 to 4 of xsin(3x) is 1/9 sin(12)-4/3 cos(12)