What is the integral of (x^3)/(csc(3x)) ?

The integral of (x^3)/(csc(3x)) is -1/3 x^3cos(3x)+1/27 (9x^2sin(3x)-2(-3xcos(3x)+sin(3x)))+C

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integral of (x^3)/(csc(3x))

\int \frac{x ^{3}}{csc ( 3 x )} d x

- \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + C

Solution steps

\int \frac{x ^{3}}{csc ( 3 x )} d x

Use the basic trigonometric identity:

\frac{1}{csc ( x )} = sin (x)

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

Apply Integration By Parts

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

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

\int - x^{2} cos (3 x) d x = - \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))

= - \frac{1}{3} x^{3} cos (3 x) - (- \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))))

Simplify

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x)))

Add a constant to the solution

= - \frac{1}{3} x^{3} cos (3 x) + \frac{1}{27} (9 x^{2} sin (3 x) - 2 (- 3 x cos (3 x) + sin (3 x))) + C

\int \frac{x}{tan ^{2} ( x )} d x

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What is the integral of (x^3)/(csc(3x)) ?
The integral of (x^3)/(csc(3x)) is -1/3 x^3cos(3x)+1/27 (9x^2sin(3x)-2(-3xcos(3x)+sin(3x)))+C