What is the integral of 15csc(3x)cot(3x) ?

The integral of 15csc(3x)cot(3x) is -5csc(3x)+C

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integral of 15csc(3x)cot(3x)

\int 15 csc (3 x) cot (3 x) d x

- 5 csc (3 x) + C

Solution steps

\int 15 csc (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

= 15 \cdot \int csc (3 x) cot (3 x) d x

Rewrite using trig identities

= 15 \cdot \int \frac{cot ( 3 x )}{sin ( 3 x )} d x

Apply u-substitution

: \int \frac{cot ( u )}{3 sin ( u )} d u

= 15 \cdot \int \frac{cot ( u )}{3 sin ( u )} d u

Take the constant out:

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

= 15 \cdot \frac{1}{3} \cdot \int \frac{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

= 15 \cdot \frac{1}{3} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= 15 \cdot \frac{1}{3} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= 15 \cdot \frac{1}{3} (- \frac{1}{v})

Substitute back

= 15 \cdot \frac{1}{3} (- \frac{1}{sin ( 3 x )})

Simplify

15 \cdot \frac{1}{3} (- \frac{1}{sin ( 3 x )}) : - 5 csc (3 x)

= - 5 csc (3 x)

Add a constant to the solution

= - 5 csc (3 x) + C

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What is the integral of 15csc(3x)cot(3x) ?
The integral of 15csc(3x)cot(3x) is -5csc(3x)+C