What is the integral of csc^2(3θ)cot(3θ) ?

The integral of csc^2(3θ)cot(3θ) is -1/6 cot^2(3θ)+C

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integral of csc^2(3θ)cot(3θ)

\int csc^{2} (3 θ) cot (3 θ) d θ

- \frac{1}{6} cot^{2} (3 θ) + C

Solution steps

\int csc^{2} (3 θ) cot (3 θ) d θ

Apply u-substitution

: \int - \frac{u}{3} d u

= \int - \frac{u}{3} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - \frac{1}{3} \cdot \frac{u ^{2}}{2}

Substitute back

u = cot (3 θ)

= - \frac{1}{3} \cdot \frac{cot ^{2} ( 3 θ )}{2}

Simplify

- \frac{1}{3} \cdot \frac{cot ^{2} ( 3 θ )}{2} : - \frac{1}{6} cot^{2} (3 θ)

= - \frac{1}{6} cot^{2} (3 θ)

Add a constant to the solution

= - \frac{1}{6} cot^{2} (3 θ) + C

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What is the integral of csc^2(3θ)cot(3θ) ?
The integral of csc^2(3θ)cot(3θ) is -1/6 cot^2(3θ)+C