What is the integral of (8x^3+4)/(sqrt(x^4+2x)) ?

The integral of (8x^3+4)/(sqrt(x^4+2x)) is 4sqrt(x^4+2x)+C

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integral of (8x^3+4)/(sqrt(x^4+2x))

\int \frac{8 x ^{3} + 4}{x ^{4} + 2 x} d x

4 x^{4} + 2 x + C

Solution steps

\int \frac{8 x ^{3} + 4}{x ^{4} + 2 x} d x

Apply u-substitution

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

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

Take the constant out:

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

= 2 \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

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

Substitute back

u = x^{4} + 2 x

= 2 \cdot 2 (x^{4} + 2 x)^{\frac{1}{2}}

Simplify

2 \cdot 2 (x^{4} + 2 x)^{\frac{1}{2}} : 4 x^{4} + 2 x

= 4 x^{4} + 2 x

Add a constant to the solution

= 4 x^{4} + 2 x + C

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What is the integral of (8x^3+4)/(sqrt(x^4+2x)) ?
The integral of (8x^3+4)/(sqrt(x^4+2x)) is 4sqrt(x^4+2x)+C