Find the values for the unknown coefficients: $A, B, C, D$. The first step is to multiply both sides of the equation from the previous step by $\left(x+3\right)^2\left(x-3\right)^2$
Expand the integral $\int\left(\frac{1}{36\left(x+3\right)^2}+\frac{1}{36\left(x-3\right)^2}+\frac{\frac{1}{126}}{x+3}+\frac{-\frac{4}{441}}{x-3}\right)dx$ into $4$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{1}{36\left(x+3\right)^2}dx$ 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 $x+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x+3$
Intermediate steps
13
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
14
Substituting $u$ and $dx$ in the integral and simplify
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.