What is the integral of csc^2(5x) ?

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

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integral of csc^2(5x)

\int csc^{2} (5 x) d x

- \frac{1}{5} cot (5 x) + C

Solution steps

\int csc^{2} (5 x) d x

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 x

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

Simplify

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

Add a constant to the solution

= - \frac{1}{5} cot (5 x) + C