What is the integral from-pi to pi of e^tcos(nt) ?

The integral from-pi to pi of e^tcos(nt) is ((e^{2pi}-1)(-1)^n)/(e^pi(n^2+1))

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integral from-pi to pi of e^tcos(nt)

\int_{- π}^{π} e^{t} cos (n t) d t

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

Solution steps

\int_{- π}^{π} e^{t} cos (n t) d t

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

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

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What is the integral from-pi to pi of e^tcos(nt) ?
The integral from-pi to pi of e^tcos(nt) is ((e^{2pi}-1)(-1)^n)/(e^pi(n^2+1))