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 $x\left(x^2+3\right)$
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
$\int\frac{2}{x}dx+2\int\frac{x}{x^2+3}dx$
12
We can solve the integral $\int\frac{x}{x^2+3}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x^2+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x^2+3$
Intermediate steps
13
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
$du=2xdx$
14
Isolate $dx$ in the previous equation
$\frac{du}{2x}=dx$
Intermediate steps
15
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int\frac{2}{x}dx$ results in: $2\ln\left(x\right)$
$2\ln\left(x\right)$
Intermediate steps
18
The integral $\int\frac{1}{u}du$ results in: $\ln\left(x^2+3\right)$
$\ln\left(x^2+3\right)$
19
Gather the results of all integrals
$2\ln\left(x\right)+\ln\left(x^2+3\right)$
20
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$2\ln\left(x\right)+\ln\left(x^2+3\right)+C_0$
Final Answer
$2\ln\left(x\right)+\ln\left(x^2+3\right)+C_0$
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The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.