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

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

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integral from-pi to pi of x^2cos(2x)

\int_{- π}^{π} x^{2} cos (2 x) d x

π

Decimal

3.14159 \dots

Solution steps

\int_{- π}^{π} x^{2} cos (2 x) d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Compute the boundaries:

π

= π

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