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

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

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integral of 1/(sin^2(ax))

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

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

Solution steps

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

Rewrite using trig identities

= \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

\int (x^{2} - 2) cos (x^{3} - 6 x) d x

\frac{\partial}{\partial x} (3 x y^{2} - 3 y^{2})

\int (x^{2} - 3 x) e^{2 x} d x

x \to 0 - lim (\frac{6 x}{x ^{4}})

y^{^{''}} - 4 y^{^{'}} - 5 y = 4 e^{- 2 t}, y (0) = 0, y^{^{'}} (0) = - 1

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