What is the integral of (x^6)/((8+x^7)^2) ?

The integral of (x^6)/((8+x^7)^2) is -1/(7(8+x^7))+C

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integral of (x^6)/((8+x^7)^2)

\int \frac{x ^{6}}{( 8 + x ^{7} ) ^{2}} d x

- \frac{1}{7 ( 8 + x ^{7} )} + C

Solution steps

\int \frac{x ^{6}}{( 8 + x ^{7} ) ^{2}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{7} (- \frac{1}{u})

Substitute back

u = 8 + x^{7}

= \frac{1}{7} (- \frac{1}{8 + x ^{7}})

Simplify

\frac{1}{7} (- \frac{1}{8 + x ^{7}}) : - \frac{1}{7 ( 8 + x ^{7} )}

= - \frac{1}{7 ( 8 + x ^{7} )}

Add a constant to the solution

= - \frac{1}{7 ( 8 + x ^{7} )} + C

f (x) = 3 x^{4} + 2 x + 8

sin (1 + x + x^{3})

x^{3} y + y^{3} x - 30

x d y = (x + y) d x

f (x) = 1 - \frac{1}{x ^{\frac{2}{3}}}

What is the integral of (x^6)/((8+x^7)^2) ?
The integral of (x^6)/((8+x^7)^2) is -1/(7(8+x^7))+C