What is the integral of ((cos(3x))^5)/(csc(3x)) ?

The integral of ((cos(3x))^5)/(csc(3x)) is -1/18 cos^6(3x)+C

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integral of ((cos(3x))^5)/(csc(3x))

\int \frac{( cos ( 3 x ) ) ^{5}}{csc ( 3 x )} d x

- \frac{1}{18} cos^{6} (3 x) + C

Solution steps

\int \frac{( cos ( 3 x ) ) ^{5}}{csc ( 3 x )} d x

Use the basic trigonometric identity:

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

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

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= - \frac{1}{15} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = cos^{5} (3 x)

= - \frac{1}{15} \cdot \frac{5}{6} (cos^{5} (3 x))^{\frac{6}{5}}

Simplify

- \frac{1}{15} \cdot \frac{5}{6} (cos^{5} (3 x))^{\frac{6}{5}} : - \frac{1}{18} cos^{6} (3 x)

= - \frac{1}{18} cos^{6} (3 x)

Add a constant to the solution

= - \frac{1}{18} cos^{6} (3 x) + C

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What is the integral of ((cos(3x))^5)/(csc(3x)) ?
The integral of ((cos(3x))^5)/(csc(3x)) is -1/18 cos^6(3x)+C