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

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

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integral of (5x^4)/(sqrt(x^5+2))

\int \frac{5 x ^{4}}{x ^{5} + 2} d x

2 x^{5} + 2 + C

Solution steps

\int \frac{5 x ^{4}}{x ^{5} + 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 ^{4}}{x ^{5} + 2} d x

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

u = x^{5} + 2

= 5 \cdot \frac{1}{5} \cdot 2 (x^{5} + 2)^{\frac{1}{2}}

Simplify

5 \cdot \frac{1}{5} \cdot 2 (x^{5} + 2)^{\frac{1}{2}} : 2 x^{5} + 2

= 2 x^{5} + 2

Add a constant to the solution

= 2 x^{5} + 2 + C

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