What is the integral of x/(sqrt(ax^2+b)) ?

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

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integral of x/(sqrt(ax^2+b))

\int \frac{x}{a x ^{2} + b} d x

\frac{a x ^{2} + b}{a} + C

Solution steps

\int \frac{x}{a x ^{2} + b} d x

Apply u-substitution

: \int \frac{1}{2 a u} d u

= \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

: 2 u^{\frac{1}{2}}

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

Substitute back

u = a x^{2} + b

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

Simplify

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

= \frac{a x ^{2} + b}{a}

Add a constant to the solution

= \frac{a x ^{2} + b}{a} + C

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