What is the integral of e^{5x}sin(7x) ?

The integral of e^{5x}sin(7x) is -(7e^{5x}cos(7x))/(74)+(5e^{5x}sin(7x))/(74)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of e^{5x}sin(7x)

\int e^{5 x} sin (7 x) d x

- \frac{7 e ^{5 x} cos ( 7 x )}{74} + \frac{5 e ^{5 x} sin ( 7 x )}{74} + C

Solution steps

\int e^{5 x} sin (7 x) d x

Apply Integration By Parts

= - \frac{1}{7} e^{5 x} cos (7 x) - \int - \frac{5}{7} e^{5 x} cos (7 x) d x

Take the constant out:

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

= - \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} \cdot \int e^{5 x} cos (7 x) d x)

Apply Integration By Parts

= - \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} (\frac{1}{7} e^{5 x} sin (7 x) - \int \frac{5}{7} e^{5 x} sin (7 x) d x))

Take the constant out:

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

= - \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} (\frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} \cdot \int e^{5 x} sin (7 x) d x))

Therefore

\int e^{5 x} sin (7 x) d x = - \frac{1}{7} e^{5 x} cos (7 x) - (- \frac{5}{7} (\frac{1}{7} e^{5 x} sin (7 x) - \frac{5}{7} \cdot \int e^{5 x} sin (7 x) d x))

Isolate

\int e^{5 x} sin (7 x) d x

= - \frac{7 e ^{5 x} cos ( 7 x )}{74} + \frac{5 e ^{5 x} sin ( 7 x )}{74}

Add a constant to the solution

= - \frac{7 e ^{5 x} cos ( 7 x )}{74} + \frac{5 e ^{5 x} sin ( 7 x )}{74} + C

What is the integral of e^{5x}sin(7x) ?
The integral of e^{5x}sin(7x) is -(7e^{5x}cos(7x))/(74)+(5e^{5x}sin(7x))/(74)+C