What is the integral of (6x^2)/(2+x^3) ?

The integral of (6x^2)/(2+x^3) is 2ln|2+x^3|+C

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integral of (6x^2)/(2+x^3)

\int \frac{6 x ^{2}}{2 + x ^{3}} d x

2 ln 2 + x^{3} + C

Solution steps

\int \frac{6 x ^{2}}{2 + x ^{3}} d x

Take the constant out:

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

= 6 \cdot \int \frac{x ^{2}}{2 + x ^{3}} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= 6 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 6 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 6 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 2 + x^{3}

= 6 \cdot \frac{1}{3} ln 2 + x^{3}

Simplify

6 \cdot \frac{1}{3} ln 2 + x^{3} : 2 ln 2 + x^{3}

= 2 ln 2 + x^{3}

Add a constant to the solution

= 2 ln 2 + x^{3} + C

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What is the integral of (6x^2)/(2+x^3) ?
The integral of (6x^2)/(2+x^3) is 2ln|2+x^3|+C