What is the integral from 0 to pi/2 of xsin(6x) ?

The integral from 0 to pi/2 of xsin(6x) is pi/(12)

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integral from 0 to pi/2 of xsin(6x)

\int_{0}^{\frac{π}{2}} x sin (6 x) d x

\frac{π}{12}

Decimal Notation

0.26179 \dots

Solution steps

\int_{0}^{\frac{π}{2}} 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}{2}}

= [\frac{1}{6} (- x cos (6 x) - 6 \cdot \int - \frac{1}{6} cos (6 x) d x)]_{0}^{\frac{π}{2}}

\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}{2}}

Simplify

[\frac{1}{6} (- x cos (6 x) - 6 (- \frac{1}{36} sin (6 x)))]_{0}^{\frac{π}{2}} : [\frac{1}{6} (- x cos (6 x) + \frac{1}{6} sin (6 x))]_{0}^{\frac{π}{2}}

= [\frac{1}{6} (- x cos (6 x) + \frac{1}{6} sin (6 x))]_{0}^{\frac{π}{2}}

Compute the boundaries:

\frac{π}{12}

= \frac{π}{12}

(x + x)

x \to \infty lim (x^{12})

\int p (p + 7)^{7} d p

\frac{\partial}{\partial x} (y^{6} sin (6 x))

\int \frac{1}{x x ^{2} + a ^{2}} d x

What is the integral from 0 to pi/2 of xsin(6x) ?
The integral from 0 to pi/2 of xsin(6x) is pi/(12)