What is the integral of x\sqrt[3]{x^2-1} ?

The integral of x\sqrt[3]{x^2-1} is 3/8 (x^2-1)^{4/3}+C

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integral of x\sqrt[3]{x^2-1}

\int x 3 x^{2} - 1 d x

\frac{3}{8} (x^{2} - 1)^{\frac{4}{3}} + C

Solution steps

\int x 3 x^{2} - 1 d x

Apply u-substitution

: \int \frac{3 u}{2} d u

= \int \frac{3 u}{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 3 u d u

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{2} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = x^{2} - 1

= \frac{1}{2} \cdot \frac{3}{4} (x^{2} - 1)^{\frac{4}{3}}

Simplify

\frac{1}{2} \cdot \frac{3}{4} (x^{2} - 1)^{\frac{4}{3}} : \frac{3}{8} (x^{2} - 1)^{\frac{4}{3}}

= \frac{3}{8} (x^{2} - 1)^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{8} (x^{2} - 1)^{\frac{4}{3}} + C

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What is the integral of x\sqrt[3]{x^2-1} ?
The integral of x\sqrt[3]{x^2-1} is 3/8 (x^2-1)^{4/3}+C