What is the integral of e^{ax}sin(2x) ?

The integral of e^{ax}sin(2x) is -(2e^{ax}cos(2x))/(4+a^2)+(ae^{ax}sin(2x))/(4+a^2)+C

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integral of e^{ax}sin(2x)

\int e^{a x} sin (2 x) d x

- \frac{2 e ^{a x} cos ( 2 x )}{4 + a ^{2}} + \frac{a e ^{a x} sin ( 2 x )}{4 + a ^{2}} + C

Solution steps

\int e^{a x} sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} e^{a x} cos (2 x) - \int - \frac{1}{2} a e^{a x} cos (2 x) d x

= - \frac{1}{2} e^{a x} cos (2 x) - \int - \frac{1}{2} a e^{a x} cos (2 x) d x

Take the constant out:

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

= - \frac{1}{2} e^{a x} cos (2 x) - (- \frac{1}{2} a \cdot \int e^{a x} cos (2 x) d x)

Simplify

- \frac{1}{2} a : - \frac{a}{2}

= - \frac{1}{2} e^{a x} cos (2 x) - (- \frac{a}{2} \cdot \int e^{a x} cos (2 x) d x)

Apply Integration By Parts

: e^{a x} \frac{1}{2} sin (2 x) - \int e^{a x} a \frac{1}{2} sin (2 x) d x

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

Take the constant out:

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

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

Simplify

a \frac{1}{2} : \frac{a}{2}

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

Therefore

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

Isolate

\int e^{a x} sin (2 x) d x

= - \frac{2 e ^{a x} cos ( 2 x )}{4 + a ^{2}} + \frac{a e ^{a x} sin ( 2 x )}{4 + a ^{2}}

Add a constant to the solution

= - \frac{2 e ^{a x} cos ( 2 x )}{4 + a ^{2}} + \frac{a e ^{a x} sin ( 2 x )}{4 + a ^{2}} + C

\int (2 x - 1) d x

\int cos (x) 3 d x

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

\int cos (x) 2 d x

\int \frac{1}{4 + e ^{x}} d x

What is the integral of e^{ax}sin(2x) ?
The integral of e^{ax}sin(2x) is -(2e^{ax}cos(2x))/(4+a^2)+(ae^{ax}sin(2x))/(4+a^2)+C