What is the integral of csc^2(2θ+9) ?

The integral of csc^2(2θ+9) is -1/2 cot(2θ+9)+C

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integral of csc^2(2θ+9)

\int csc^{2} (2 θ + 9) d θ

- \frac{1}{2} cot (2 θ + 9) + C

Solution steps

\int csc^{2} (2 θ + 9) d θ

Rewrite using trig identities

= \int \frac{1}{sin ^{2} ( 2 θ + 9 )} d θ

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{sin ^{2} ( u )} d u = - cot (u)

= \frac{1}{2} (- cot (u))

Substitute back

u = 2 θ + 9

= \frac{1}{2} (- cot (2 θ + 9))

Simplify

= - \frac{1}{2} cot (2 θ + 9)

Add a constant to the solution

= - \frac{1}{2} cot (2 θ + 9) + C

What is the integral of csc^2(2θ+9) ?
The integral of csc^2(2θ+9) is -1/2 cot(2θ+9)+C