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

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

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integral of ((2x^2)/((3-x^3)^2))

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

\frac{2}{3 ( 3 - x ^{3} )} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{3 u ^{2}} d u

= 2 \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

= 2 (- \frac{1}{3} \cdot \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

: - \frac{1}{u}

= 2 (- \frac{1}{3} (- \frac{1}{u}))

Substitute back

u = 3 - x^{3}

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

Simplify

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

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

Add a constant to the solution

= \frac{2}{3 ( 3 - x ^{3} )} + C

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