What is the integral of x(x^2+3)^{-12/7} ?

The integral of x(x^2+3)^{-12/7} is -7/(10(x^2+3)^{5/7)}+C

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integral of x(x^2+3)^{-12/7}

\int x (x^{2} + 3)^{- \frac{12}{7}} d x

- \frac{7}{10 ( x ^{2} + 3 ) ^{\frac{5}{7}}} + C

Solution steps

\int x (x^{2} + 3)^{- \frac{12}{7}} d x

Apply u-substitution

: \int \frac{1}{2 u ^{\frac{12}{7}}} d u

= \int \frac{1}{2 u ^{\frac{12}{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 ^{\frac{12}{7}}} d u

Apply the Power Rule

: - \frac{7}{5 u ^{\frac{5}{7}}}

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

Substitute back

u = x^{2} + 3

= \frac{1}{2} (- \frac{7}{5 ( x ^{2} + 3 ) ^{\frac{5}{7}}})

Simplify

\frac{1}{2} (- \frac{7}{5 ( x ^{2} + 3 ) ^{\frac{5}{7}}}) : - \frac{7}{10 ( x ^{2} + 3 ) ^{\frac{5}{7}}}

= - \frac{7}{10 ( x ^{2} + 3 ) ^{\frac{5}{7}}}

Add a constant to the solution

= - \frac{7}{10 ( x ^{2} + 3 ) ^{\frac{5}{7}}} + C

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What is the integral of x(x^2+3)^{-12/7} ?
The integral of x(x^2+3)^{-12/7} is -7/(10(x^2+3)^{5/7)}+C