What is the integral of 2x^3sin(5x) ?

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

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integral of 2x^3sin(5x)

\int 2 x^{3} sin (5 x) d x

2 (- \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 2 x^{3} sin (5 x) d x

Take the constant out:

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

= 2 \cdot \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

= 2 (- \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)))

= 2 (- \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

= 2 (- \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

= 2 (- \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

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