Final Answer
Step-by-step Solution
Specify the solving method
Rewrite the expression $\frac{x^2-9x+15}{x^3-6x^2+12x-8}$ inside the integral in factored form
Rewrite the fraction $\frac{x^2-9x+15}{\left(x-2\right)^{3}}$ in $3$ simpler fractions using partial fraction decomposition
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 $\left(x-2\right)^{3}$
Multiplying polynomials
Simplifying
Assigning values to $x$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{x^2-9x+15}{\left(x-2\right)^{3}}$ in decomposed fraction equals
Expand the integral $\int\left(\frac{1}{x-2}+\frac{-5}{\left(x-2\right)^{2}}+\frac{1}{\left(x-2\right)^{3}}\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{1}{x-2}dx$ 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 $x-2$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int\frac{1}{u}du$ results in: $\ln\left(x-2\right)$
Gather the results of all integrals
We can solve the integral $\int\frac{-5}{\left(x-2\right)^{2}}dx$ 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 $x-2$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int\frac{-5}{u^{2}}du$ results in: $\frac{5}{x-2}$
Gather the results of all integrals
We can solve the integral $\int\frac{1}{\left(x-2\right)^{3}}dx$ 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 $x-2$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int\frac{1}{u^{3}}du$ results in: $\frac{1}{-2\left(x-2\right)^{2}}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$