What is the integral of (x^4+1)/(sqrt(x^5+5x)) ?

The integral of (x^4+1)/(sqrt(x^5+5x)) is 2/5 sqrt(x^5+5x)+C

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integral of (x^4+1)/(sqrt(x^5+5x))

\int \frac{x ^{4} + 1}{x ^{5} + 5 x} d x

\frac{2}{5} x^{5} + 5 x + C

Solution steps

\int \frac{x ^{4} + 1}{x ^{5} + 5 x} d x

Apply u-substitution

= \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

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

Apply the Power Rule

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

Substitute back

u = x^{5} + 5 x

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

Simplify

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

= \frac{2}{5} x^{5} + 5 x

Add a constant to the solution

= \frac{2}{5} x^{5} + 5 x + C

What is the integral of (x^4+1)/(sqrt(x^5+5x)) ?
The integral of (x^4+1)/(sqrt(x^5+5x)) is 2/5 sqrt(x^5+5x)+C