What is the integral of 1/(sqrt(x)sin^2(\sqrt{x))} ?

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

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integral of 1/(sqrt(x)sin^2(\sqrt{x))}

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

- 2 cot (x) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 (- cot (u))

Substitute back

u = x

= 2 (- cot (x))

Simplify

= - 2 cot (x)

Add a constant to the solution

= - 2 cot (x) + C

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