What is the integral of (2x^2)/(x^3+8) ?

The integral of (2x^2)/(x^3+8) is 2/3 ln|x^3+8|+C

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integral of (2x^2)/(x^3+8)

\int \frac{2 x ^{2}}{x ^{3} + 8} d x

\frac{2}{3} ln x^{3} + 8 + C

Solution steps

\int \frac{2 x ^{2}}{x ^{3} + 8} 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}}{x ^{3} + 8} d x

Apply u-substitution

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

= 2 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = x^{3} + 8

= 2 \cdot \frac{1}{3} ln x^{3} + 8

Simplify

2 \cdot \frac{1}{3} ln x^{3} + 8 : \frac{2}{3} ln x^{3} + 8

= \frac{2}{3} ln x^{3} + 8

Add a constant to the solution

= \frac{2}{3} ln x^{3} + 8 + C

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What is the integral of (2x^2)/(x^3+8) ?
The integral of (2x^2)/(x^3+8) is 2/3 ln|x^3+8|+C