What is the integral of (3x^2+6)/(x^3+6x) ?

The integral of (3x^2+6)/(x^3+6x) is ln|x|+ln|x^2+6|+C

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integral of (3x^2+6)/(x^3+6x)

\int \frac{3 x ^{2} + 6}{x ^{3} + 6 x} d x

ln ∣ x ∣ + ln x^{2} + 6 + C

Solution steps

\int \frac{3 x ^{2} + 6}{x ^{3} + 6 x} d x

Take the partial fraction of

\frac{3 x ^{2} + 6}{x ^{3} + 6 x} : \frac{1}{x} + \frac{2 x}{x ^{2} + 6}

= \int \frac{1}{x} + \frac{2 x}{x ^{2} + 6} 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} + 6} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2 x}{x ^{2} + 6} d x = ln x^{2} + 6

= ln ∣ x ∣ + ln x^{2} + 6

Add a constant to the solution

= ln ∣ x ∣ + ln x^{2} + 6 + C

\int_{0}^{1} (2 x - 1) e^{2 x} d x

\int 45 sin (3) (x) cos (2) (x) d x

x^{3}

x \to 1 + lim (e^{x})

x^{\frac{4}{7}}

What is the integral of (3x^2+6)/(x^3+6x) ?
The integral of (3x^2+6)/(x^3+6x) is ln|x|+ln|x^2+6|+C