What is the integral of (x^7)/((4-x^8)^2) ?

The integral of (x^7)/((4-x^8)^2) is 1/(8(4-x^8))+C

integral of (x^7)/((4-x^8)^2)

\int \frac{x ^{7}}{( 4 - x ^{8} ) ^{2}} d x

\frac{1}{8 ( 4 - x ^{8} )} + C

Solution steps

\int \frac{x ^{7}}{( 4 - x ^{8} ) ^{2}} d x

Apply u-substitution

= \int - \frac{1}{8 u ^{2}} d u

Take the constant out:

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

= - \frac{1}{8} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

= - \frac{1}{8} (- \frac{1}{u})

Substitute back

u = 4 - x^{8}

= - \frac{1}{8} (- \frac{1}{4 - x ^{8}})

Simplify

- \frac{1}{8} (- \frac{1}{4 - x ^{8}}) : \frac{1}{8 ( 4 - x ^{8} )}

= \frac{1}{8 ( 4 - x ^{8} )}

Add a constant to the solution

= \frac{1}{8 ( 4 - x ^{8} )} + C

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What is the integral of (x^7)/((4-x^8)^2) ?
The integral of (x^7)/((4-x^8)^2) is 1/(8(4-x^8))+C