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

The integral of e^{kx}sin(x) is -(e^{kx}cos(x))/(1+k^2)+(ke^{kx}sin(x))/(1+k^2)+C

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integral of e^{kx}sin(x)

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

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

Solution steps

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

Apply Integration By Parts

: - e^{k x} cos (x) - \int - k e^{k x} cos (x) d x

= - e^{k x} cos (x) - \int - k e^{k x} cos (x) d x

Take the constant out:

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

= - e^{k x} cos (x) - (- k \cdot \int e^{k x} cos (x) d x)

Apply Integration By Parts

: e^{k x} sin (x) - \int e^{k x} k sin (x) d x

= - e^{k x} cos (x) - (- k (e^{k x} sin (x) - \int e^{k x} k sin (x) d x))

Take the constant out:

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

= - e^{k x} cos (x) - (- k (e^{k x} sin (x) - k \cdot \int e^{k x} sin (x) d x))

Therefore

\int e^{k x} sin (x) d x = - e^{k x} cos (x) - (- k (e^{k x} sin (x) - k \cdot \int e^{k x} sin (x) d x))

Isolate

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

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

Add a constant to the solution

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

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What is the integral of e^{kx}sin(x) ?
The integral of e^{kx}sin(x) is -(e^{kx}cos(x))/(1+k^2)+(ke^{kx}sin(x))/(1+k^2)+C