What is the integral from 0 to pi/3 of xsin(2x) ?

The integral from 0 to pi/3 of xsin(2x) is (2pi+3sqrt(3))/(24)

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integral from 0 to pi/3 of xsin(2x)

\int_{0}^{\frac{π}{3}} x sin (2 x) d x

\frac{2 π + 3 3}{24}

Decimal

0.47830 \dots

Solution steps

\int_{0}^{\frac{π}{3}} x sin (2 x) d x

Apply Integration By Parts

= [\frac{1}{2} (- x cos (2 x) - 2 \cdot \int - \frac{1}{2} cos (2 x) d x)]_{0}^{\frac{π}{3}}

\int - \frac{1}{2} cos (2 x) d x = - \frac{1}{4} sin (2 x)

= [\frac{1}{2} (- x cos (2 x) - 2 (- \frac{1}{4} sin (2 x)))]_{0}^{\frac{π}{3}}

Simplify

[\frac{1}{2} (- x cos (2 x) - 2 (- \frac{1}{4} sin (2 x)))]_{0}^{\frac{π}{3}} : [\frac{1}{2} (- x cos (2 x) + \frac{1}{2} sin (2 x))]_{0}^{\frac{π}{3}}

= [\frac{1}{2} (- x cos (2 x) + \frac{1}{2} sin (2 x))]_{0}^{\frac{π}{3}}

Compute the boundaries:

\frac{2 π + 3 3}{24}

= \frac{2 π + 3 3}{24}

What is the integral from 0 to pi/3 of xsin(2x) ?
The integral from 0 to pi/3 of xsin(2x) is (2pi+3sqrt(3))/(24)