What is the integral of x^3sin(5x) ?

The integral of x^3sin(5x) is -1/5 x^3cos(5x)+3/625 (25x^2sin(5x)-2(-5xcos(5x)+sin(5x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x^3sin(5x)

\int x^{3} sin (5 x) d x

- \frac{1}{5} x^{3} cos (5 x) + \frac{3}{625} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + C

Solution steps

\int x^{3} sin (5 x) d x

Apply Integration By Parts

= - \frac{1}{5} x^{3} cos (5 x) - \int - \frac{3}{5} x^{2} cos (5 x) d x

\int - \frac{3}{5} x^{2} cos (5 x) d x = - \frac{3}{625} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

= - \frac{1}{5} x^{3} cos (5 x) - (- \frac{3}{625} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))))

Simplify

= - \frac{1}{5} x^{3} cos (5 x) + \frac{3}{625} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

Add a constant to the solution

= - \frac{1}{5} x^{3} cos (5 x) + \frac{3}{625} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + C

What is the integral of x^3sin(5x) ?
The integral of x^3sin(5x) is -1/5 x^3cos(5x)+3/625 (25x^2sin(5x)-2(-5xcos(5x)+sin(5x)))+C