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

The integral of cos(x)e^{-3x} is (e^{-3x}sin(x))/(10)-(3e^{-3x}cos(x))/(10)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cos(x)e^{-3x}

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

\frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10} + C

Solution steps

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

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= \frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10}

Add a constant to the solution

= \frac{e ^{- 3 x} sin ( x )}{10} - \frac{3 e ^{- 3 x} cos ( x )}{10} + C

What is the integral of cos(x)e^{-3x} ?
The integral of cos(x)e^{-3x} is (e^{-3x}sin(x))/(10)-(3e^{-3x}cos(x))/(10)+C