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

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

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

integral

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

\frac{1}{3} ln x^{3} + 2 + C

Solution steps

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

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = x^{3} + 2

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

Add a constant to the solution

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

\frac{2 x}{( x + 1 ) ^{2} ( x ^{2} + 2 )}

\frac{sin ( x )}{( 1 + ( cos ( x ) ) ^{2} )}

\frac{7 x ^{3} - 5 x ^{2} + 7 x - 8}{x ^{2}}

2 (x + 1)

- 2 x

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