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

Popular >

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

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

= \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

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

= 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

\int 7 x (x^{2} + 6)^{2} d x

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

csc (x)

\int \frac{x + 2}{2 x} d x

\int 6 sec^{2} (6 x) tan (6 x) d x

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