What is the integral of (3x^2)/((x^3+1)^3) ?

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

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integral of (3x^2)/((x^3+1)^3)

\int \frac{3 x ^{2}}{( x ^{3} + 1 ) ^{3}} d x

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

Solution steps

\int \frac{3 x ^{2}}{( x ^{3} + 1 ) ^{3}} d x

Take the constant out:

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

= 3 \cdot \int \frac{x ^{2}}{( x ^{3} + 1 ) ^{3}} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = x^{3} + 1

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

Simplify

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

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

Add a constant to the solution

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

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