What is the integral of csc^2(ax) ?

The integral of csc^2(ax) is -1/a cot(ax)+C

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integral of csc^2(ax)

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = a x

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

Simplify

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

Add a constant to the solution

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

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