Final Answer
$2\ln\left(x\right)-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$
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Step-by-step Solution
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Intermediate steps
1
Rewrite the expression $\frac{4x^2+6}{x^3+3x}$ inside the integral in factored form
$\int\frac{4x^2+6}{x\left(x^2+3\right)}dx$
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2
Rewrite the fraction $\frac{4x^2+6}{x\left(x^2+3\right)}$ in $2$ simpler fractions using partial fraction decomposition
$\frac{4x^2+6}{x\left(x^2+3\right)}=\frac{A}{x}+\frac{Bx+C}{x^2+3}$
3
Find the values for the unknown coefficients: $A, B, C$. The first step is to multiply both sides of the equation from the previous step by $x\left(x^2+3\right)$
$4x^2+6=x\left(x^2+3\right)\left(\frac{A}{x}+\frac{Bx+C}{x^2+3}\right)$
4
Multiplying polynomials
$4x^2+6=\frac{x\left(x^2+3\right)A}{x}+\frac{x\left(x^2+3\right)\left(Bx+C\right)}{x^2+3}$
$4x^2+6=\left(x^2+3\right)A+x\left(Bx+C\right)$
6
Assigning values to $x$ we obtain the following system of equations
$\begin{matrix}6=3A&\:\:\:\:\:\:\:(x=0) \\ 42=12A+9B-3C&\:\:\:\:\:\:\:(x=-3) \\ 42=12A+9B+3C&\:\:\:\:\:\:\:(x=3)\end{matrix}$
7
Proceed to solve the system of linear equations
$\begin{matrix}3A & + & 0B & + & 0C & =6 \\ 12A & + & 9B & - & 3C & =42 \\ 12A & + & 9B & + & 3C & =42\end{matrix}$
8
Rewrite as a coefficient matrix
$\left(\begin{matrix}3 & 0 & 0 & 6 \\ 12 & 9 & -3 & 42 \\ 12 & 9 & 3 & 42\end{matrix}\right)$
9
Reducing the original matrix to a identity matrix using Gaussian Elimination
$\left(\begin{matrix}1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0\end{matrix}\right)$
10
The integral of $\frac{4x^2+6}{x\left(x^2+3\right)}$ in decomposed fraction equals
$\int\left(\frac{2}{x}+\frac{2x}{x^2+3}\right)dx$
Intermediate steps
11
Simplify the expression inside the integral
$\int\frac{2}{x}dx+2\int\frac{x}{x^2+3}dx$
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Intermediate steps
12
The integral $\int\frac{2}{x}dx$ results in: $2\ln\left(x\right)$
$2\ln\left(x\right)$
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Intermediate steps
13
The integral $2\int\frac{x}{x^2+3}dx$ results in: $-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)$
$-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)$
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14
Gather the results of all integrals
$2\ln\left(x\right)-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)$
15
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$2\ln\left(x\right)-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$
Final Answer$2\ln\left(x\right)-2\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$