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

The integral of x/(sin^2(2x^2)) is -1/4 cot(2x^2)+C

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integral of x/(sin^2(2x^2))

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

- \frac{1}{4} cot (2 x^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{4 sin ^{2} ( u )} d u

= \int \frac{1}{4 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}{4} \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}{4} (- cot (u))

Substitute back

u = 2 x^{2}

= \frac{1}{4} (- cot (2 x^{2}))

Simplify

= - \frac{1}{4} cot (2 x^{2})

Add a constant to the solution

= - \frac{1}{4} cot (2 x^{2}) + C

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What is the integral of x/(sin^2(2x^2)) ?
The integral of x/(sin^2(2x^2)) is -1/4 cot(2x^2)+C