What is the integral of (x^2)/(5x^3+3) ?

The integral of (x^2)/(5x^3+3) is 1/15 ln|5x^3+3|+C

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integral of (x^2)/(5x^3+3)

\int \frac{x ^{2}}{5 x ^{3} + 3} d x

\frac{1}{15} ln 5 x^{3} + 3 + C

Solution steps

\int \frac{x ^{2}}{5 x ^{3} + 3} d x

Apply u-substitution

= \int \frac{1}{15 u} d u

Take the constant out:

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

= \frac{1}{15} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{15} ln ∣ u ∣

Substitute back

u = 5 x^{3} + 3

= \frac{1}{15} ln 5 x^{3} + 3

Add a constant to the solution

= \frac{1}{15} ln 5 x^{3} + 3 + C

What is the integral of (x^2)/(5x^3+3) ?
The integral of (x^2)/(5x^3+3) is 1/15 ln|5x^3+3|+C