What is the integral of (7x^6)/((4+x^7)^5) ?

The integral of (7x^6)/((4+x^7)^5) is -1/(4(4+x^7)^4)+C

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integral of (7x^6)/((4+x^7)^5)

\int \frac{7 x ^{6}}{( 4 + x ^{7} ) ^{5}} d x

- \frac{1}{4 ( 4 + x ^{7} ) ^{4}} + C

Solution steps

\int \frac{7 x ^{6}}{( 4 + x ^{7} ) ^{5}} d x

Take the constant out:

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

= 7 \cdot \int \frac{x ^{6}}{( 4 + x ^{7} ) ^{5}} d x

Apply u-substitution

: \int \frac{1}{7 u ^{5}} d u

= 7 \cdot \int \frac{1}{7 u ^{5}} d u

Take the constant out:

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

= 7 \cdot \frac{1}{7} \cdot \int \frac{1}{u ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

= 7 \cdot \frac{1}{7} (- \frac{1}{4 u ^{4}})

Substitute back

u = 4 + x^{7}

= 7 \cdot \frac{1}{7} (- \frac{1}{4 ( 4 + x ^{7} ) ^{4}})

Simplify

7 \cdot \frac{1}{7} (- \frac{1}{4 ( 4 + x ^{7} ) ^{4}}) : - \frac{1}{4 ( 4 + x ^{7} ) ^{4}}

= - \frac{1}{4 ( 4 + x ^{7} ) ^{4}}

Add a constant to the solution

= - \frac{1}{4 ( 4 + x ^{7} ) ^{4}} + C

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What is the integral of (7x^6)/((4+x^7)^5) ?
The integral of (7x^6)/((4+x^7)^5) is -1/(4(4+x^7)^4)+C