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

The integral of e^{ax}sin(x) is -(e^{ax}cos(x))/(1+a^2)+(ae^{ax}sin(x))/(1+a^2)+C

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integral of e^{ax}sin(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int a^{3 x} d x

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\int \frac{- 1}{x ( ln ( x ) ) ^{3}} d x

What is the integral of e^{ax}sin(x) ?
The integral of e^{ax}sin(x) is -(e^{ax}cos(x))/(1+a^2)+(ae^{ax}sin(x))/(1+a^2)+C