What is the integral of e^{nx}sin(nx) ?

The integral of e^{nx}sin(nx) is -(e^{nx}cos(nx))/(2n)+(e^{nx}sin(nx))/(2n)+C

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integral of e^{nx}sin(nx)

\int e^{n x} sin (n x) d x

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

Solution steps

\int e^{n x} sin (n x) d x

Apply Integration By Parts

: - e^{n x} \frac{1}{n} cos (n x) - \int - e^{n x} cos (n x) d x

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

: e^{n x} \frac{1}{n} sin (n x) - \int e^{n x} sin (n x) d x

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

Therefore

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

Isolate

\int e^{n x} sin (n x) d x

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

Add a constant to the solution

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

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What is the integral of e^{nx}sin(nx) ?
The integral of e^{nx}sin(nx) is -(e^{nx}cos(nx))/(2n)+(e^{nx}sin(nx))/(2n)+C