What is the integral of sin(2x)e^{3x} ?

The integral of sin(2x)e^{3x} is -(2e^{3x}cos(2x))/(13)+(3e^{3x}sin(2x))/(13)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(2x)e^{3x}

\int sin (2 x) e^{3 x} d x

- \frac{2 e ^{3 x} cos ( 2 x )}{13} + \frac{3 e ^{3 x} sin ( 2 x )}{13} + C

Solution steps

\int sin (2 x) e^{3 x} d x

Apply Integration By Parts

: - \frac{1}{2} e^{3 x} cos (2 x) - \int - \frac{3}{2} e^{3 x} cos (2 x) d x

= - \frac{1}{2} e^{3 x} cos (2 x) - \int - \frac{3}{2} e^{3 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^{3 x} cos (2 x) - (- \frac{3}{2} \cdot \int e^{3 x} cos (2 x) d x)

Apply Integration By Parts

: \frac{1}{2} e^{3 x} sin (2 x) - \int \frac{3}{2} e^{3 x} sin (2 x) d x

= - \frac{1}{2} e^{3 x} cos (2 x) - (- \frac{3}{2} (\frac{1}{2} e^{3 x} sin (2 x) - \int \frac{3}{2} e^{3 x} 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^{3 x} cos (2 x) - (- \frac{3}{2} (\frac{1}{2} e^{3 x} sin (2 x) - \frac{3}{2} \cdot \int e^{3 x} sin (2 x) d x))

Therefore

\int e^{3 x} sin (2 x) d x = - \frac{1}{2} e^{3 x} cos (2 x) - (- \frac{3}{2} (\frac{1}{2} e^{3 x} sin (2 x) - \frac{3}{2} \cdot \int e^{3 x} sin (2 x) d x))

Isolate

\int e^{3 x} sin (2 x) d x

= - \frac{2 e ^{3 x} cos ( 2 x )}{13} + \frac{3 e ^{3 x} sin ( 2 x )}{13}

Add a constant to the solution

= - \frac{2 e ^{3 x} cos ( 2 x )}{13} + \frac{3 e ^{3 x} sin ( 2 x )}{13} + C

\int 24 x^{5} arccot (2 x^{3}) d x

\int \frac{1}{x ^{2} ( x ^{2} + 4 )} d x

\int \frac{x ^{3} + 1}{x ^{3} + x} d x

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

What is the integral of sin(2x)e^{3x} ?
The integral of sin(2x)e^{3x} is -(2e^{3x}cos(2x))/(13)+(3e^{3x}sin(2x))/(13)+C