What is the integral of (24x)/(\sqrt[3]{6x^2+1)} ?

The integral of (24x)/(\sqrt[3]{6x^2+1)} is 3(6x^2+1)^{2/3}+C

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integral of (24x)/(\sqrt[3]{6x^2+1)}

\int \frac{24 x}{3 6 x ^{2} + 1} d x

3 (6 x^{2} + 1)^{\frac{2}{3}} + C

Solution steps

\int \frac{24 x}{3 6 x ^{2} + 1} d x

Take the constant out:

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

= 24 \cdot \int \frac{x}{3 6 x ^{2} + 1} d x

Apply u-substitution

: \int \frac{1}{2 3 6 u + 1} d u

= 24 \cdot \int \frac{1}{2 3 6 u + 1} d u

Take the constant out:

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

= 24 \cdot \frac{1}{2} \cdot \int \frac{1}{3 6 u + 1} d u

Apply u-substitution

: \int \frac{1}{6 3 v} d v

= 24 \cdot \frac{1}{2} \cdot \int \frac{1}{6 3 v} d v

Take the constant out:

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

= 24 \cdot \frac{1}{2} \cdot \frac{1}{6} \cdot \int \frac{1}{3 v} d v

Apply the Power Rule

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

= 24 \cdot \frac{1}{2} \cdot \frac{1}{6} \cdot \frac{3}{2} v^{\frac{2}{3}}

Substitute back

= 24 \cdot \frac{1}{2} \cdot \frac{1}{6} \cdot \frac{3}{2} (6 x^{2} + 1)^{\frac{2}{3}}

Simplify

24 \cdot \frac{1}{2} \cdot \frac{1}{6} \cdot \frac{3}{2} (6 x^{2} + 1)^{\frac{2}{3}} : 3 (6 x^{2} + 1)^{\frac{2}{3}}

= 3 (6 x^{2} + 1)^{\frac{2}{3}}

Add a constant to the solution

= 3 (6 x^{2} + 1)^{\frac{2}{3}} + C

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What is the integral of (24x)/(\sqrt[3]{6x^2+1)} ?
The integral of (24x)/(\sqrt[3]{6x^2+1)} is 3(6x^2+1)^{2/3}+C