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

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

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integral from 0 to pi of 4xsin(2x)

\int_{0}^{π} 4 x sin (2 x) d x

- 2 π

Decimal

- 6.28318 \dots

Solution steps

\int_{0}^{π} 4 x sin (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int_{0}^{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}

= 4 [\frac{1}{2} (- x cos (2 x) - 2 \cdot \int - \frac{1}{2} cos (2 x) d x)]_{0}^{π}

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

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

Simplify

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

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

Compute the boundaries:

- \frac{π}{2}

= 4 (- \frac{π}{2})

Simplify

= - 2 π

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