What is the integral from 0 to pi/6 of xsin(6x) ?

The integral from 0 to pi/6 of xsin(6x) is pi/(36)

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integral from 0 to pi/6 of xsin(6x)

\int_{0}^{\frac{π}{6}} x sin (6 x) d x

\frac{π}{36}

Decimal Notation

0.08726 \dots

Solution steps

\int_{0}^{\frac{π}{6}} x sin (6 x) d x

Apply Integration By Parts

: [\frac{1}{6} (- x cos (6 x) - 6 \cdot \int - \frac{1}{6} cos (6 x) d x)]_{0}^{\frac{π}{6}}

= [\frac{1}{6} (- x cos (6 x) - 6 \cdot \int - \frac{1}{6} cos (6 x) d x)]_{0}^{\frac{π}{6}}

\int - \frac{1}{6} cos (6 x) d x = - \frac{1}{36} sin (6 x)

= [\frac{1}{6} (- x cos (6 x) - 6 (- \frac{1}{36} sin (6 x)))]_{0}^{\frac{π}{6}}

Simplify

[\frac{1}{6} (- x cos (6 x) - 6 (- \frac{1}{36} sin (6 x)))]_{0}^{\frac{π}{6}} : [\frac{1}{6} (- x cos (6 x) + \frac{1}{6} sin (6 x))]_{0}^{\frac{π}{6}}

= [\frac{1}{6} (- x cos (6 x) + \frac{1}{6} sin (6 x))]_{0}^{\frac{π}{6}}

Compute the boundaries:

\frac{π}{36}

= \frac{π}{36}

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What is the integral from 0 to pi/6 of xsin(6x) ?
The integral from 0 to pi/6 of xsin(6x) is pi/(36)