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

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

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integral of (2x^3+x)/(\sqrt[3]{x^4+x^2+3)}

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

Solution steps

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

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

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

Substitute back

u = x^{4} + x^{2} + 3

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

Simplify

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

= \frac{3}{4} (x^{4} + x^{2} + 3)^{\frac{2}{3}}

Add a constant to the solution

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

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