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

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

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

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

ln x^{3} + 5 x + 2 + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x^{3} + 5 x + 2

= ln x^{3} + 5 x + 2

Add a constant to the solution

= ln x^{3} + 5 x + 2 + C

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