What is the integral of (8x^2)/(sqrt(x^3+2)) ?

The integral of (8x^2)/(sqrt(x^3+2)) is 16/3 sqrt(x^3+2)+C

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integral of (8x^2)/(sqrt(x^3+2))

\int \frac{8 x ^{2}}{x ^{3} + 2} d x

\frac{16}{3} x^{3} + 2 + C

Solution steps

\int \frac{8 x ^{2}}{x ^{3} + 2} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{2}}{x ^{3} + 2} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= 8 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 8 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 8 \cdot \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{3} + 2

= 8 \cdot \frac{1}{3} \cdot 2 (x^{3} + 2)^{\frac{1}{2}}

Simplify

8 \cdot \frac{1}{3} \cdot 2 (x^{3} + 2)^{\frac{1}{2}} : \frac{16}{3} x^{3} + 2

= \frac{16}{3} x^{3} + 2

Add a constant to the solution

= \frac{16}{3} x^{3} + 2 + C

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What is the integral of (8x^2)/(sqrt(x^3+2)) ?
The integral of (8x^2)/(sqrt(x^3+2)) is 16/3 sqrt(x^3+2)+C