What is the integral of (5x^4)/((x^5+1)^2) ?

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

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integral of (5x^4)/((x^5+1)^2)

\int \frac{5 x ^{4}}{( x ^{5} + 1 ) ^{2}} d x

- \frac{1}{x ^{5} + 1} + C

Solution steps

\int \frac{5 x ^{4}}{( x ^{5} + 1 ) ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{4}}{( x ^{5} + 1 ) ^{2}} d x

Apply u-substitution

= 5 \cdot \int \frac{1}{5 u ^{2}} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

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

Substitute back

u = x^{5} + 1

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

Simplify

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

= - \frac{1}{x ^{5} + 1}

Add a constant to the solution

= - \frac{1}{x ^{5} + 1} + C

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