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

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

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

\int_{0}^{π} e^{x} sin (n x) d x

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

Solution steps

\int_{0}^{π} e^{x} sin (n x) d x

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

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

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

\int_{0}^{5} (x + 2) x + 4 d x

\int_{\frac{1}{x}}^{x} \frac{x ^{2}}{y ^{2}} d y

\int_{0}^{2} x^{3} + 1 d x

\int_{100}^{200} 0.1 e^{- 0.002 x} d x

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