What is the integral from 0 to pi of e^{ax}sin(nx) ?

The integral from 0 to pi of e^{ax}sin(nx) is (-(-1)^nne^{pia}+n)/(a^2+n^2)

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integral from 0 to pi of e^{ax}sin(nx)

\int_{0}^{π} e^{a x} sin (n x) d x

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

Solution steps

\int_{0}^{π} e^{a x} sin (n x) d x

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

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

\int_{0}^{3} x \cdot e^{x} d x

\int_{0}^{\frac{π}{4}} e^{x} cos (x) d x

\int_{0}^{3} \frac{5}{x ^{2} + 1} d x

\int_{0}^{1} 2 π (x + 2) x^{2} d x

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

What is the integral from 0 to pi of e^{ax}sin(nx) ?
The integral from 0 to pi of e^{ax}sin(nx) is (-(-1)^nne^{pia}+n)/(a^2+n^2)