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

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

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

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

ln ∣ x ∣ + ln x^{2} + 8 + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x} + \frac{2 x}{x ^{2} + 8} 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{1}{x} d x + \int \frac{2 x}{x ^{2} + 8} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2 x}{x ^{2} + 8} d x = ln x^{2} + 8

= ln ∣ x ∣ + ln x^{2} + 8

Add a constant to the solution

= ln ∣ x ∣ + ln x^{2} + 8 + C

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