What is the integral of 6csc^2(3x)cot(3x) ?

The integral of 6csc^2(3x)cot(3x) is -cot^2(3x)+C

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integral of 6csc^2(3x)cot(3x)

\int 6 csc^{2} (3 x) cot (3 x) d x

- cot^{2} (3 x) + C

Solution steps

\int 6 csc^{2} (3 x) cot (3 x) d x

Take the constant out:

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

= 6 \cdot \int csc^{2} (3 x) cot (3 x) d x

Apply u-substitution

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

= 6 \cdot \int - \frac{u}{3} d u

Take the constant out:

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

= 6 (- \frac{1}{3} \cdot \int u d u)

Apply the Power Rule

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

= 6 (- \frac{1}{3} \cdot \frac{u ^{2}}{2})

Substitute back

u = cot (3 x)

= 6 (- \frac{1}{3} \cdot \frac{cot ^{2} ( 3 x )}{2})

Simplify

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

= - cot^{2} (3 x)

Add a constant to the solution

= - cot^{2} (3 x) + C

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