What is the integral of 1/((sin(3x))^2) ?

The integral of 1/((sin(3x))^2) is -1/3 cot(3x)+C

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integral of 1/((sin(3x))^2)

\int \frac{1}{( sin ( 3 x ) ) ^{2}} d x

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

Solution steps

\int \frac{1}{( sin ( 3 x ) ) ^{2}} d x

Use the basic trigonometric identity:

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

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

Apply u-substitution

: \int csc^{2} (u) \frac{1}{3} d u

= \int csc^{2} (u) \frac{1}{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 csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{3} (- cot (u))

Substitute back

u = 3 x

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

Simplify

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

Add a constant to the solution

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

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What is the integral of 1/((sin(3x))^2) ?
The integral of 1/((sin(3x))^2) is -1/3 cot(3x)+C