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
Rewrite the fraction $\frac{x}{x^2+3}$ inside the integral as the product of two functions: $x\frac{1}{x^2+3}$
$\int\frac{2}{x}dx+2\int x\frac{1}{x^2+3}dx$
13
We can solve the integral $\int x\frac{1}{x^2+3}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Multiply the single term $2$ by each term of the polynomial $\left(\frac{\sqrt{3}}{3}x\arctan\left(\frac{x}{\sqrt{3}}\right)-\frac{\sqrt{3}}{3}\int\arctan\left(\frac{x}{\sqrt{3}}\right)dx\right)$
The integral $\int\frac{2}{x}dx$ results in: $2\ln\left(x\right)$
$2\ln\left(x\right)$
Intermediate steps
21
The integral $-\frac{2\sqrt{3}}{3}\int\arctan\left(\frac{x}{\sqrt{3}}\right)dx$ results in: $-\frac{2\sqrt{3}}{3}x\arctan\left(\frac{x}{\sqrt{3}}\right)+\ln\left(x^2+3\right)$
Cancel like terms $\frac{2\sqrt{3}}{3}x\arctan\left(\frac{x}{\sqrt{3}}\right)$ and $-\frac{2\sqrt{3}}{3}x\arctan\left(\frac{x}{\sqrt{3}}\right)$
$2\ln\left(x\right)+\ln\left(x^2+3\right)$
24
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.