What is the integral of e^{-3x}sin(6x) ?

The integral of e^{-3x}sin(6x) is -(2e^{-3x}cos(6x))/(15)-(e^{-3x}sin(6x))/(15)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{-3x}sin(6x)

\int e^{- 3 x} sin (6 x) d x

- \frac{2 e ^{- 3 x} cos ( 6 x )}{15} - \frac{e ^{- 3 x} sin ( 6 x )}{15} + C

Solution steps

\int e^{- 3 x} sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} e^{- 3 x} cos (6 x) - \int \frac{1}{2} e^{- 3 x} cos (6 x) d x

= - \frac{1}{6} e^{- 3 x} cos (6 x) - \int \frac{1}{2} e^{- 3 x} cos (6 x) d x

Take the constant out:

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

= - \frac{1}{6} e^{- 3 x} cos (6 x) - \frac{1}{2} \cdot \int e^{- 3 x} cos (6 x) d x

Apply Integration By Parts

: \frac{1}{6} e^{- 3 x} sin (6 x) - \int - \frac{1}{2} e^{- 3 x} sin (6 x) d x

= - \frac{1}{6} e^{- 3 x} cos (6 x) - \frac{1}{2} (\frac{1}{6} e^{- 3 x} sin (6 x) - \int - \frac{1}{2} e^{- 3 x} sin (6 x) d x)

Take the constant out:

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

= - \frac{1}{6} e^{- 3 x} cos (6 x) - \frac{1}{2} (\frac{1}{6} e^{- 3 x} sin (6 x) - (- \frac{1}{2} \cdot \int e^{- 3 x} sin (6 x) d x))

Therefore

\int e^{- 3 x} sin (6 x) d x = - \frac{1}{6} e^{- 3 x} cos (6 x) - \frac{1}{2} (\frac{1}{6} e^{- 3 x} sin (6 x) - (- \frac{1}{2} \cdot \int e^{- 3 x} sin (6 x) d x))

Isolate

\int e^{- 3 x} sin (6 x) d x

= - \frac{2 e ^{- 3 x} cos ( 6 x )}{15} - \frac{e ^{- 3 x} sin ( 6 x )}{15}

Add a constant to the solution

= - \frac{2 e ^{- 3 x} cos ( 6 x )}{15} - \frac{e ^{- 3 x} sin ( 6 x )}{15} + C

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

\int \frac{x + 1}{( x ^{2} - 4 x + 6 ) ^{2}} d x

\int \frac{( ln ( x ) ) ^{2}}{x [ ( ln ( x ) ) ^{3} + 8 ]} d x

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

\int \frac{4}{3} x^{\frac{1}{3}} d x

What is the integral of e^{-3x}sin(6x) ?
The integral of e^{-3x}sin(6x) is -(2e^{-3x}cos(6x))/(15)-(e^{-3x}sin(6x))/(15)+C