What is the integral of (x^5)/((x^6+1)^2) ?

The integral of (x^5)/((x^6+1)^2) is -1/(6(x^6+1))+C

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integral of (x^5)/((x^6+1)^2)

\int \frac{x ^{5}}{( x ^{6} + 1 ) ^{2}} d x

- \frac{1}{6 ( x ^{6} + 1 )} + C

Solution steps

\int \frac{x ^{5}}{( x ^{6} + 1 ) ^{2}} d x

Apply u-substitution

: \int \frac{1}{6 u ^{2}} d u

= \int \frac{1}{6 u ^{2}} 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 ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{6} (- \frac{1}{u})

Substitute back

u = x^{6} + 1

= \frac{1}{6} (- \frac{1}{x ^{6} + 1})

Simplify

\frac{1}{6} (- \frac{1}{x ^{6} + 1}) : - \frac{1}{6 ( x ^{6} + 1 )}

= - \frac{1}{6 ( x ^{6} + 1 )}

Add a constant to the solution

= - \frac{1}{6 ( x ^{6} + 1 )} + C

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What is the integral of (x^5)/((x^6+1)^2) ?
The integral of (x^5)/((x^6+1)^2) is -1/(6(x^6+1))+C