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

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

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

integral

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

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

Solution steps

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

Apply u-substitution

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

= \int \frac{1}{2 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}{2} \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}{2} (- cot (u))

Substitute back

u = x^{2}

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

Simplify

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

Add a constant to the solution

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

\frac{x}{16}

\frac{cos ^{2} ( 2 x )}{sin ( 2 x )}

- 3 sin (x)

\frac{( x + 1 ) ^{3}}{x ^{2}}

\frac{2 x + 6}{x ^{2} + 6 x - 3}

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