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

The integral of csc(3x)cot(3x) is -1/3 csc(3x)+C

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

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

- \frac{1}{3} csc (3 x) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \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

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{v}

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

Substitute back

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

Simplify

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

= - \frac{1}{3} csc (3 x)

Add a constant to the solution

= - \frac{1}{3} csc (3 x) + C

\int (3 y^{2}) d y

y = x + x, (49, 56)

\int_{1}^{25} \frac{2}{x} d x

x \to + 0 lim (\frac{x ^{2}}{1 - cos ( x )})

\frac{d y}{d x} = \frac{5 x}{7 y}

What is the integral of csc(3x)cot(3x) ?
The integral of csc(3x)cot(3x) is -1/3 csc(3x)+C