What is the integral of (x^4)/((2+x^5)^6) ?

The integral of (x^4)/((2+x^5)^6) is -1/(25(2+x^5)^5)+C

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integral of (x^4)/((2+x^5)^6)

\int \frac{x ^{4}}{( 2 + x ^{5} ) ^{6}} d x

- \frac{1}{25 ( 2 + x ^{5} ) ^{5}} + C

Solution steps

\int \frac{x ^{4}}{( 2 + x ^{5} ) ^{6}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{5 u ^{5}}

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

Substitute back

u = 2 + x^{5}

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

Simplify

\frac{1}{5} (- \frac{1}{5 ( 2 + x ^{5} ) ^{5}}) : - \frac{1}{25 ( 2 + x ^{5} ) ^{5}}

= - \frac{1}{25 ( 2 + x ^{5} ) ^{5}}

Add a constant to the solution

= - \frac{1}{25 ( 2 + x ^{5} ) ^{5}} + C

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What is the integral of (x^4)/((2+x^5)^6) ?
The integral of (x^4)/((2+x^5)^6) is -1/(25(2+x^5)^5)+C