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

The integral from 0 to pi of e^xcos(nx) is (e^pi(-1)^n-1)/(n^2+1)

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integral from 0 to pi of e^xcos(nx)

\int_{0}^{π} e^{x} cos (n x) d x

\frac{e ^{π} ( - 1 ) ^{n} - 1}{n ^{2} + 1}

Solution steps

\int_{0}^{π} e^{x} cos (n x) d x

Compute the indefinite integral:

\int e^{x} cos (n x) d x = \frac{n e ^{x} sin ( n x )}{n ^{2} + 1} + \frac{e ^{x} cos ( n x )}{n ^{2} + 1} + C

Compute the boundaries:

\int_{0}^{π} e^{x} cos (n x) d x = \frac{e ^{π} ( - 1 ) ^{n}}{n ^{2} + 1} - \frac{1}{n ^{2} + 1}

= \frac{e ^{π} ( - 1 ) ^{n}}{n ^{2} + 1} - \frac{1}{n ^{2} + 1}

Simplify

= \frac{e ^{π} ( - 1 ) ^{n} - 1}{n ^{2} + 1}

\int_{2}^{3} x - 1 d x

\int_{0}^{π} sin (x) - 6 d x

\int_{π}^{0} sin^{2} (x) d x

\int_{0}^{3} 2 x e^{- x} d x

What is the integral from 0 to pi of e^xcos(nx) ?
The integral from 0 to pi of e^xcos(nx) is (e^pi(-1)^n-1)/(n^2+1)