What is the integral of x/((x^2-5)^{1/7)} ?

The integral of x/((x^2-5)^{1/7)} is 7/12 (x^2-5)^{6/7}+C

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integral of x/((x^2-5)^{1/7)}

\int \frac{x}{( x ^{2} - 5 ) ^{\frac{1}{7}}} d x

\frac{7}{12} (x^{2} - 5)^{\frac{6}{7}} + C

Solution steps

\int \frac{x}{( x ^{2} - 5 ) ^{\frac{1}{7}}} d x

Apply u-substitution

: \int \frac{1}{2 ( u - 5 ) ^{\frac{1}{7}}} d u

= \int \frac{1}{2 ( u - 5 ) ^{\frac{1}{7}}} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{( u - 5 ) ^{\frac{1}{7}}} d u

Apply u-substitution

: \int \frac{1}{v ^{\frac{1}{7}}} d v

= \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{1}{7}}} d v

Apply the Power Rule

: \frac{7}{6} v^{\frac{6}{7}}

= \frac{1}{2} \cdot \frac{7}{6} v^{\frac{6}{7}}

Substitute back

= \frac{1}{2} \cdot \frac{7}{6} (x^{2} - 5)^{\frac{6}{7}}

Simplify

\frac{1}{2} \cdot \frac{7}{6} (x^{2} - 5)^{\frac{6}{7}} : \frac{7}{12} (x^{2} - 5)^{\frac{6}{7}}

= \frac{7}{12} (x^{2} - 5)^{\frac{6}{7}}

Add a constant to the solution

= \frac{7}{12} (x^{2} - 5)^{\frac{6}{7}} + C

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What is the integral of x/((x^2-5)^{1/7)} ?
The integral of x/((x^2-5)^{1/7)} is 7/12 (x^2-5)^{6/7}+C