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

The integral of ((x^4))/((x^5+x)) is 1/4 ln|x^4+1|+C

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

integral

\frac{( x ^{4} )}{( x ^{5} + x )}

\frac{1}{4} ln x^{4} + 1 + C

Solution steps

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

Simplify

\frac{x ^{4}}{x ^{5} + x} : \frac{x ^{3}}{x ^{4} + 1}

= \int \frac{x ^{3}}{x ^{4} + 1} d x

Apply u-substitution

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 1

= \frac{1}{4} ln x^{4} + 1

Add a constant to the solution

= \frac{1}{4} ln x^{4} + 1 + C

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