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

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

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integral of e^{ax}sin(nx)

\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

Solution steps

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

Apply Integration By Parts

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

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

Simplify

- a \frac{1}{n} : - \frac{a}{n}

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

Apply Integration By Parts

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

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

Simplify

a \frac{1}{n} : \frac{a}{n}

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

Therefore

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

Isolate

\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}}

Add a constant to the solution

= - \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

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