What is the integral of csc^2(5θ) ?

The integral of csc^2(5θ) is -1/5 cot(5θ)+C

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integral of csc^2(5θ)

\int csc^{2} (5 θ) d θ

- \frac{1}{5} cot (5 θ) + C

Solution steps

\int csc^{2} (5 θ) d θ

Apply u-substitution

= \int csc^{2} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int csc^{2} (u) d u

Use the common integral:

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

= \frac{1}{5} (- cot (u))

Substitute back

u = 5 θ

= \frac{1}{5} (- cot (5 θ))

Simplify

= - \frac{1}{5} cot (5 θ)

Add a constant to the solution

= - \frac{1}{5} cot (5 θ) + C