What is the integral of xsqrt(x^2-a) ?

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

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integral of xsqrt(x^2-a)

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

\frac{1}{3} (x^{2} - a)^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int v d v

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{3} (x^{2} - a)^{\frac{3}{2}} + C

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