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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int - \frac{1}{2 a u} d u

Take the constant out:

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

= - \frac{1}{2 a} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= - \frac{1}{2 a} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 - a x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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