What is the integral of (5x^2)/(sqrt(2x^3-4)) ?

The integral of (5x^2)/(sqrt(2x^3-4)) is 5/3 sqrt(2x^3-4)+C

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integral of (5x^2)/(sqrt(2x^3-4))

\int \frac{5 x ^{2}}{2 x ^{3} - 4} d x

\frac{5}{3} 2 x^{3} - 4 + C

Solution steps

\int \frac{5 x ^{2}}{2 x ^{3} - 4} 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}}{2 x ^{3} - 4} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 5 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{6} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 x^{3} - 4

= 5 \cdot \frac{1}{6} \cdot 2 (2 x^{3} - 4)^{\frac{1}{2}}

Simplify

5 \cdot \frac{1}{6} \cdot 2 (2 x^{3} - 4)^{\frac{1}{2}} : \frac{5}{3} 2 x^{3} - 4

= \frac{5}{3} 2 x^{3} - 4

Add a constant to the solution

= \frac{5}{3} 2 x^{3} - 4 + C

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What is the integral of (5x^2)/(sqrt(2x^3-4)) ?
The integral of (5x^2)/(sqrt(2x^3-4)) is 5/3 sqrt(2x^3-4)+C