What is the integral of (x^5)/(1+x^6) ?

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

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integral of (x^5)/(1+x^6)

\int \frac{x ^{5}}{1 + x ^{6}} d x

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

Solution steps

\int \frac{x ^{5}}{1 + x ^{6}} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= \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 = 1 + x^{6}

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

Add a constant to the solution

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

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