We can solve the integral $\int\frac{3x+5}{\left(x-1\right)^2\left(x+1\right)}dx$ 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
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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
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Rewriting $x$ in terms of $u$
$x=u+1$
Intermediate steps
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Substituting $u$, $dx$ and $x$ in the integral and simplify
$\int\frac{8+3u}{u^2\left(2+u\right)}du$
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Rewrite the fraction $\frac{8+3u}{u^2\left(2+u\right)}$ 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 $u^2\left(2+u\right)$
Expand the integral $\int\left(\frac{4}{u^2}+\frac{1}{2\left(2+u\right)}+\frac{-1}{2u}\right)du$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
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.