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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int cot (u) csc (2 u) d u

= \int cot (u) csc (2 u) d u

Rewrite using trig identities

= \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 = arcsin (\frac{1}{2} x)

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

Simplify

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

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

Add a constant to the solution

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

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