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

The integral of (x^2)/((ax^3+b)^2) is -1/(3a(ax^3+b))+C

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integral of (x^2)/((ax^3+b)^2)

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

- \frac{1}{3 a ( a x ^{3} + b )} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = a x^{3} + b

= \frac{1}{3 a} (- \frac{1}{a x ^{3} + b})

Simplify

\frac{1}{3 a} (- \frac{1}{a x ^{3} + b}) : - \frac{1}{3 a ( a x ^{3} + b )}

= - \frac{1}{3 a ( a x ^{3} + b )}

Add a constant to the solution

= - \frac{1}{3 a ( a x ^{3} + b )} + C

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What is the integral of (x^2)/((ax^3+b)^2) ?
The integral of (x^2)/((ax^3+b)^2) is -1/(3a(ax^3+b))+C