What is the integral from 0 to pi of sin^2(x)e^x ?

The integral from 0 to pi of sin^2(x)e^x is (2e^pi-2)/5

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integral from 0 to pi of sin^2(x)e^x

\int_{0}^{π} sin^{2} (x) e^{x} d x

\frac{2 e ^{π} - 2}{5}

Decimal

8.85627 \dots

Solution steps

\int_{0}^{π} sin^{2} (x) e^{x} d x

Apply Integration By Parts

: [sin^{2} (x) e^{x} - \int sin (2 x) e^{x} d x]_{0}^{π}

= [sin^{2} (x) e^{x} - \int sin (2 x) e^{x} d x]_{0}^{π}

\int sin (2 x) e^{x} d x = - \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5}

= [sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5})]_{0}^{π}

Simplify

[sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5})]_{0}^{π} : [\frac{1}{5} (5 e^{x} sin^{2} (x) + 2 e^{x} cos (2 x) - e^{x} sin (2 x))]_{0}^{π}

= [\frac{1}{5} (5 e^{x} sin^{2} (x) + 2 e^{x} cos (2 x) - e^{x} sin (2 x))]_{0}^{π}

Compute the boundaries:

\frac{2 e ^{π}}{5} - \frac{2}{5}

= \frac{2 e ^{π}}{5} - \frac{2}{5}

Simplify

= \frac{2 e ^{π} - 2}{5}

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What is the integral from 0 to pi of sin^2(x)e^x ?
The integral from 0 to pi of sin^2(x)e^x is (2e^pi-2)/5