What is the integral of 1/(x^2*sqrt(1-x^2)) ?

The integral of 1/(x^2*sqrt(1-x^2)) is -(sqrt(1-x^2))/x+C

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integral of 1/(x^2*sqrt(1-x^2))

\int \frac{1}{x ^{2} \cdot 1 - x ^{2}} d x

- \frac{1 - x ^{2}}{x} + C

Solution steps

\int \frac{1}{x ^{2} 1 - x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

= \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

Simplify

\frac{cot ( u )}{cos ( u ) sin ( u )} : csc^{2} (u)

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

Use the common integral:

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

= - cot (u)

Substitute back

u = arcsin (x)

= - cot (arcsin (x))

Simplify

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

= - \frac{1 - x ^{2}}{x}

Add a constant to the solution

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

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