What is the integral of (x^5)/((x^6+3)^{5/6)} ?

The integral of (x^5)/((x^6+3)^{5/6)} is (x^6+3)^{1/6}+C

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integral of (x^5)/((x^6+3)^{5/6)}

\int \frac{x ^{5}}{( x ^{6} + 3 ) ^{\frac{5}{6}}} d x

(x^{6} + 3)^{\frac{1}{6}} + C

Solution steps

\int \frac{x ^{5}}{( x ^{6} + 3 ) ^{\frac{5}{6}}} d x

Apply u-substitution

: \int \frac{1}{6 u ^{\frac{5}{6}}} d u

= \int \frac{1}{6 u ^{\frac{5}{6}}} 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 ^{\frac{5}{6}}} d u

Apply the Power Rule

: 6 u^{\frac{1}{6}}

= \frac{1}{6} \cdot 6 u^{\frac{1}{6}}

Substitute back

u = x^{6} + 3

= \frac{1}{6} \cdot 6 (x^{6} + 3)^{\frac{1}{6}}

Simplify

\frac{1}{6} \cdot 6 (x^{6} + 3)^{\frac{1}{6}} : (x^{6} + 3)^{\frac{1}{6}}

= (x^{6} + 3)^{\frac{1}{6}}

Add a constant to the solution

= (x^{6} + 3)^{\frac{1}{6}} + C

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What is the integral of (x^5)/((x^6+3)^{5/6)} ?
The integral of (x^5)/((x^6+3)^{5/6)} is (x^6+3)^{1/6}+C