What is the integral of (2x^2)/(sqrt(x^3+12)) ?

The integral of (2x^2)/(sqrt(x^3+12)) is 4/3 sqrt(x^3+12)+C

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integral of (2x^2)/(sqrt(x^3+12))

\int \frac{2 x ^{2}}{x ^{3} + 12} d x

\frac{4}{3} x^{3} + 12 + C

Solution steps

\int \frac{2 x ^{2}}{x ^{3} + 12} d x

Take the constant out:

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

= 2 \cdot \int \frac{x ^{2}}{x ^{3} + 12} d x

Apply u-substitution

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

= 2 \cdot \int \frac{2}{3} d u

Integral of a constant:

\int a d x = a x

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

Substitute back

u = x^{3} + 12

= 2 \cdot \frac{2}{3} x^{3} + 12

Simplify

2 \cdot \frac{2}{3} x^{3} + 12 : \frac{4}{3} x^{3} + 12

= \frac{4}{3} x^{3} + 12

Add a constant to the solution

= \frac{4}{3} x^{3} + 12 + C

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What is the integral of (2x^2)/(sqrt(x^3+12)) ?
The integral of (2x^2)/(sqrt(x^3+12)) is 4/3 sqrt(x^3+12)+C