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

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

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integral from-pi to pi of xsin(2x)

\int_{- π}^{π} x sin (2 x) d x

- π

Decimal

- 3.14159 \dots

Solution steps

\int_{- π}^{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)]_{- π}^{π}

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

\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)))]_{- π}^{π}

Simplify

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

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

Compute the boundaries:

- π

= - π

\int_{1}^{4} 5 x^{2} d x

\int_{0}^{0.5} \frac{sin ( 3 x )}{x} d x

\int_{1}^{e} x e^{2 x^{2} - 1} d x

\int_{1}^{6} x^{2} - 2 x - 8 d x

\int_{1}^{4} x^{2} + 4 x + 5 d x

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