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 $x\left(x^2-2x-1\right)$
Expand the integral $\int\left(\frac{1}{x}+\frac{-x+6}{x^2-2x-1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
$\int\frac{1}{x}dx+\int\frac{-x+6}{x^2-2x-1}dx$
Intermediate steps
12
The integral $\int\frac{1}{x}dx$ results in: $\ln\left(x\right)$
$\ln\left(x\right)$
13
Gather the results of all integrals
$\ln\left(x\right)+\int\frac{-x+6}{x^2-2x-1}dx$
Intermediate steps
14
Rewrite the expression $\frac{-x+6}{x^2-2x-1}$ inside the integral in factored form
The integral $\int\frac{-x+6}{-2+\left(x-1\right)^2}dx$ results in: $-\frac{5\sqrt{2}}{4}\ln\left(0.414214+x\right)+\frac{5\sqrt{2}}{4}\ln\left(-2.414214+x\right)+\ln\left(\frac{\sqrt{2}}{\sqrt{-2+\left(x-1\right)^2}}\right)$
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.