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

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

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

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

ln x^{2} + 3 + 2 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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