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
We can solve the integral $\int\frac{-x+6}{-2+\left(x-1\right)^2}dx$ 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 $x-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x-1$
Intermediate steps
16
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
$du=dx$
Intermediate steps
17
Rewriting $x$ in terms of $u$
$x=u+1$
Intermediate steps
18
Substituting $u$, $dx$ and $x$ in the integral and simplify
$\ln\left(x\right)+\int\frac{5-u}{-2+u^2}du$
Intermediate steps
19
The integral $\int\frac{5-u}{-2+u^2}du$ results in: $-\frac{5\sqrt{2}}{2}\ln\left(\frac{x-1}{\sqrt{-2+\left(x-1\right)^2}}+\frac{\sqrt{2}}{\sqrt{-2+\left(x-1\right)^2}}\right)-\frac{1}{2}\ln\left(-2+\left(x-1\right)^2\right)$
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
$L.C.M.=\sqrt{-2+\left(x-1\right)^2}$
Intermediate steps
22
Combine and simplify all terms in the same fraction with common denominator $\sqrt{-2+\left(x-1\right)^2}$
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