What is the integral of x^3sin(7x) ?

The integral of x^3sin(7x) is -1/7 x^3cos(7x)+3/2401 (49x^2sin(7x)-2(-7xcos(7x)+sin(7x)))+C

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integral of x^3sin(7x)

\int x^{3} sin (7 x) d x

- \frac{1}{7} x^{3} cos (7 x) + \frac{3}{2401} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + C

Solution steps

\int x^{3} sin (7 x) d x

Apply Integration By Parts

: - \frac{1}{7} x^{3} cos (7 x) - \int - \frac{3}{7} x^{2} cos (7 x) d x

= - \frac{1}{7} x^{3} cos (7 x) - \int - \frac{3}{7} x^{2} cos (7 x) d x

\int - \frac{3}{7} x^{2} cos (7 x) d x = - \frac{3}{2401} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

= - \frac{1}{7} x^{3} cos (7 x) - (- \frac{3}{2401} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))))

Simplify

= - \frac{1}{7} x^{3} cos (7 x) + \frac{3}{2401} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

Add a constant to the solution

= - \frac{1}{7} x^{3} cos (7 x) + \frac{3}{2401} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + C

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What is the integral of x^3sin(7x) ?
The integral of x^3sin(7x) is -1/7 x^3cos(7x)+3/2401 (49x^2sin(7x)-2(-7xcos(7x)+sin(7x)))+C