What is the integral of (4x^2)/((4+x^3)^2) ?

The integral of (4x^2)/((4+x^3)^2) is -4/(3(4+x^3))+C

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integral of (4x^2)/((4+x^3)^2)

\int \frac{4 x ^{2}}{( 4 + x ^{3} ) ^{2}} d x

- \frac{4}{3 ( 4 + x ^{3} )} + C

Solution steps

\int \frac{4 x ^{2}}{( 4 + x ^{3} ) ^{2}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x ^{2}}{( 4 + x ^{3} ) ^{2}} d x

Apply u-substitution

= 4 \cdot \int \frac{1}{3 u ^{2}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

= 4 \cdot \frac{1}{3} (- \frac{1}{u})

Substitute back

u = 4 + x^{3}

= 4 \cdot \frac{1}{3} (- \frac{1}{4 + x ^{3}})

Simplify

4 \cdot \frac{1}{3} (- \frac{1}{4 + x ^{3}}) : - \frac{4}{3 ( 4 + x ^{3} )}

= - \frac{4}{3 ( 4 + x ^{3} )}

Add a constant to the solution

= - \frac{4}{3 ( 4 + x ^{3} )} + C

What is the integral of (4x^2)/((4+x^3)^2) ?
The integral of (4x^2)/((4+x^3)^2) is -4/(3(4+x^3))+C