What is the integral of 1/(x^2(a^2-x^2)^{1/2)} ?

The integral of 1/(x^2(a^2-x^2)^{1/2)} is -1/(ax)+C

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integral of 1/(x^2(a^2-x^2)^{1/2)}

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

- \frac{1}{a x} + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

= \int x^{- 2} \frac{1}{( a ^{2} ) ^{\frac{1}{2}}} d x

Take the constant out:

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

= \frac{1}{( a ^{2} ) ^{\frac{1}{2}}} \cdot \int x^{- 2} d x

Apply the Power Rule

: - \frac{1}{x}

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

Simplify

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

= - \frac{1}{a x}

Add a constant to the solution

= - \frac{1}{a x} + C

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What is the integral of 1/(x^2(a^2-x^2)^{1/2)} ?
The integral of 1/(x^2(a^2-x^2)^{1/2)} is -1/(ax)+C