What is the integral of (e^x)*sin(nx) ?

The integral of (e^x)*sin(nx) is -(ne^xcos(nx))/(n^2+1)+(e^xsin(nx))/(n^2+1)+C

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integral of (e^x)*sin(nx)

\int (e^{x}) \cdot 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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

\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}

Add a constant to the solution

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

\int \frac{cos ( ln ( 4 x ) )}{x} d x

\int (2 x + 8) d x

\int \frac{2 x + 3}{x ( x - 1 )} d x

What is the integral of (e^x)*sin(nx) ?
The integral of (e^x)*sin(nx) is -(ne^xcos(nx))/(n^2+1)+(e^xsin(nx))/(n^2+1)+C