What is the integral of (5x^2)/(sqrt(x^3-2)) ?

The integral of (5x^2)/(sqrt(x^3-2)) is 10/3 sqrt(x^3-2)+C

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integral of (5x^2)/(sqrt(x^3-2))

\int \frac{5 x ^{2}}{x ^{3} - 2} d x

\frac{10}{3} x^{3} - 2 + C

Solution steps

\int \frac{5 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

= 5 \cdot \int \frac{x ^{2}}{x ^{3} - 2} d x

Apply u-substitution

: \int \frac{u}{2 x u ^{\frac{3}{2}} - 2} d u

= 5 \cdot \int \frac{1}{3 u - 2} d u

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \int \frac{1}{u - 2} d u

Apply u-substitution

: \int 2 d v

= 5 \cdot \frac{1}{3} \cdot \int 2 d v

Integral of a constant:

\int a d x = a x

= 5 \cdot \frac{1}{3} \cdot 2 v

Substitute back

= 5 \cdot \frac{1}{3} \cdot 2 x^{3} - 2

Simplify

5 \cdot \frac{1}{3} \cdot 2 x^{3} - 2 : \frac{10}{3} x^{3} - 2

= \frac{10}{3} x^{3} - 2

Add a constant to the solution

= \frac{10}{3} x^{3} - 2 + C

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What is the integral of (5x^2)/(sqrt(x^3-2)) ?
The integral of (5x^2)/(sqrt(x^3-2)) is 10/3 sqrt(x^3-2)+C