What is the integral of (4x^2+6)/(x(x^2+3)) ?

The integral of (4x^2+6)/(x(x^2+3)) is ln|x^2+3|+2ln|x|+C

integral of (4x^2+6)/(x(x^2+3))

\int \frac{4 x ^{2} + 6}{x ( x ^{2} + 3 )} d x

ln x^{2} + 3 + 2 ln ∣ x ∣ + C

Solution steps

\int \frac{4 x ^{2} + 6}{x ( x ^{2} + 3 )} d x

Expand

\frac{4 x ^{2} + 6}{x ( x ^{2} + 3 )} : \frac{4 x ^{2}}{x ( x ^{2} + 3 )} + \frac{6}{x ( x ^{2} + 3 )}

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

\int \frac{4 x ^{2}}{x ( x ^{2} + 3 )} d x = 2 ln x^{2} + 3

\int \frac{6}{x ( x ^{2} + 3 )} d x = 2 ln ∣ x ∣ - ln x^{2} + 3

= 2 ln x^{2} + 3 + 2 ln ∣ x ∣ - ln x^{2} + 3

Simplify

= ln x^{2} + 3 + 2 ln ∣ x ∣

Add a constant to the solution

= ln x^{2} + 3 + 2 ln ∣ x ∣ + C

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What is the integral of (4x^2+6)/(x(x^2+3)) ?
The integral of (4x^2+6)/(x(x^2+3)) is ln|x^2+3|+2ln|x|+C