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

The integral from 0 to pi of e^xcos^2(x) is (3e^pi-3)/5

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

\int_{0}^{π} e^{x} cos^{2} (x) d x

\frac{3 e ^{π} - 3}{5}

Decimal

13.28441 \dots

Solution steps

\int_{0}^{π} e^{x} cos^{2} (x) d x

Apply Integration By Parts

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

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

\int - e^{x} sin (2 x) d x = \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x)

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

Simplify

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

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

Compute the boundaries:

\frac{3 e ^{π}}{5} - \frac{3}{5}

= \frac{3 e ^{π}}{5} - \frac{3}{5}

Simplify

= \frac{3 e ^{π} - 3}{5}

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What is the integral from 0 to pi of e^xcos^2(x) ?
The integral from 0 to pi of e^xcos^2(x) is (3e^pi-3)/5