What is the integral of e^{3x+1}cos(5x+3) ?

The integral of e^{3x+1}cos(5x+3) is (5e^{3x+1}sin(5x+3))/(34)+(3e^{3x+1}cos(5x+3))/(34)+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+1}cos(5x+3)

\int e^{3 x + 1} cos (5 x + 3) d x

\frac{5 e ^{3 x + 1} sin ( 5 x + 3 )}{34} + \frac{3 e ^{3 x + 1} cos ( 5 x + 3 )}{34} + C

Solution steps

\int e^{3 x + 1} cos (5 x + 3) d x

Apply Integration By Parts

: \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \int \frac{3}{5} e^{3 x + 1} sin (5 x + 3) d x

= \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \int \frac{3}{5} e^{3 x + 1} sin (5 x + 3) d x

Take the constant out:

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

= \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \frac{3}{5} \cdot \int e^{3 x + 1} sin (5 x + 3) d x

Apply Integration By Parts

: - \frac{1}{5} e^{3 x + 1} cos (5 x + 3) - \int - \frac{3}{5} e^{3 x + 1} cos (5 x + 3) d x

= \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \frac{3}{5} (- \frac{1}{5} e^{3 x + 1} cos (5 x + 3) - \int - \frac{3}{5} e^{3 x + 1} cos (5 x + 3) d x)

Take the constant out:

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

= \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \frac{3}{5} (- \frac{1}{5} e^{3 x + 1} cos (5 x + 3) - (- \frac{3}{5} \cdot \int e^{3 x + 1} cos (5 x + 3) d x))

Therefore

\int e^{3 x + 1} cos (5 x + 3) d x = \frac{1}{5} e^{3 x + 1} sin (5 x + 3) - \frac{3}{5} (- \frac{1}{5} e^{3 x + 1} cos (5 x + 3) - (- \frac{3}{5} \cdot \int e^{3 x + 1} cos (5 x + 3) d x))

Isolate

\int e^{3 x + 1} cos (5 x + 3) d x

= \frac{5 e ^{3 x + 1} sin ( 5 x + 3 )}{34} + \frac{3 e ^{3 x + 1} cos ( 5 x + 3 )}{34}

Add a constant to the solution

= \frac{5 e ^{3 x + 1} sin ( 5 x + 3 )}{34} + \frac{3 e ^{3 x + 1} cos ( 5 x + 3 )}{34} + C

\int \frac{3 ln ( x ) + 1}{x} d x

\int \frac{8}{x ^{2} + 14 x + 53} d x

\int \frac{1}{cos ( x )} d cos (x) d x

\int 6 x^{2} + 5 d x

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

What is the integral of e^{3x+1}cos(5x+3) ?
The integral of e^{3x+1}cos(5x+3) is (5e^{3x+1}sin(5x+3))/(34)+(3e^{3x+1}cos(5x+3))/(34)+C