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

The integral of (x^2+3)/(x^3+2x) is 3/2 ln|x|-1/4 ln|x^2+2|+C

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

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

\frac{3}{2} ln ∣ x ∣ - \frac{1}{4} ln x^{2} + 2 + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{3}{2 x} - \frac{x}{2 ( x ^{2} + 2 )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{3}{2 x} d x - \int \frac{x}{2 ( x ^{2} + 2 )} d x

\int \frac{3}{2 x} d x = \frac{3}{2} ln ∣ x ∣

\int \frac{x}{2 ( x ^{2} + 2 )} d x = \frac{1}{4} ln x^{2} + 2

= \frac{3}{2} ln ∣ x ∣ - \frac{1}{4} ln x^{2} + 2

Add a constant to the solution

= \frac{3}{2} ln ∣ x ∣ - \frac{1}{4} ln x^{2} + 2 + C

What is the integral of (x^2+3)/(x^3+2x) ?
The integral of (x^2+3)/(x^3+2x) is 3/2 ln|x|-1/4 ln|x^2+2|+C