What is the integral of (x^5)/(x^6+5) ?

The integral of (x^5)/(x^6+5) is 1/6 ln|x^6+5|+C

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integral of (x^5)/(x^6+5)

\int \frac{x ^{5}}{x ^{6} + 5} d x

\frac{1}{6} ln x^{6} + 5 + C

Solution steps

\int \frac{x ^{5}}{x ^{6} + 5} d x

Apply u-substitution

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = x^{6} + 5

= \frac{1}{6} ln x^{6} + 5

Add a constant to the solution

= \frac{1}{6} ln x^{6} + 5 + C

What is the integral of (x^5)/(x^6+5) ?
The integral of (x^5)/(x^6+5) is 1/6 ln|x^6+5|+C