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 $2\int\frac{x}{x^2+3}dx$ by applying integration method of trigonometric substitution using the substitution
$x=\sqrt{3}\tan\left(\theta \right)$
Intermediate steps
13
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
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.