What is the integral of xsqrt(ax^2+b) ?

The integral of xsqrt(ax^2+b) is ((ax^2+b)^{3/2})/(3a)+C

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integral of xsqrt(ax^2+b)

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

\frac{( a x ^{2} + b ) ^{\frac{3}{2}}}{3 a} + C

Solution steps

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

Apply u-substitution

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

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

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{2 a} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = a x^{2} + b

= \frac{1}{2 a} \cdot \frac{2}{3} (a x^{2} + b)^{\frac{3}{2}}

Simplify

\frac{1}{2 a} \cdot \frac{2}{3} (a x^{2} + b)^{\frac{3}{2}} : \frac{( a x ^{2} + b ) ^{\frac{3}{2}}}{3 a}

= \frac{( a x ^{2} + b ) ^{\frac{3}{2}}}{3 a}

Add a constant to the solution

= \frac{( a x ^{2} + b ) ^{\frac{3}{2}}}{3 a} + C

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What is the integral of xsqrt(ax^2+b) ?
The integral of xsqrt(ax^2+b) is ((ax^2+b)^{3/2})/(3a)+C