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

The integral from 0 to 3/2 pi of xsin(2x) is (3pi)/4

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

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

\frac{3 π}{4}

Decimal

2.35619 \dots

Solution steps

\int_{0}^{\frac{3}{2} π} 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}^{π \frac{3}{2}}

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

\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}^{π \frac{3}{2}}

Simplify

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

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

Compute the boundaries:

\frac{3 π}{4}

= \frac{3 π}{4}

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\int_{1}^{2} (x^{3} + 3) d x

What is the integral from 0 to 3/2 pi of xsin(2x) ?
The integral from 0 to 3/2 pi of xsin(2x) is (3pi)/4