What is the integral of (1-a^2x^2)^{(-3)/2} ?

The integral of (1-a^2x^2)^{(-3)/2} is x/((1-a^2x^2)^{1/2)}+C

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integral of (1-a^2x^2)^{(-3)/2}

\int (1 - a^{2} x^{2})^{\frac{- 3}{2}} d x

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

Solution steps

\int (1 - a^{2} x^{2})^{\frac{- 3}{2}} d x

Apply Trigonometric Substitution

= \int \frac{1}{a cos ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{cos ^{2} ( u )} d u

Use the common integral:

\int \frac{1}{cos ^{2} ( u )} d u = tan (u)

= \frac{1}{a} tan (u)

Substitute back

u = arcsin (a x)

= \frac{1}{a} tan (arcsin (a x))

Simplify

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

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

Add a constant to the solution

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

What is the integral of (1-a^2x^2)^{(-3)/2} ?
The integral of (1-a^2x^2)^{(-3)/2} is x/((1-a^2x^2)^{1/2)}+C