Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
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
Take the constant $\frac{1}{36}$ out of the integral
$\frac{1}{36}\int\frac{1}{\left(x+3\right)^2}dx$
Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(c-1\right)\left(x+a\right)^{\left(c-1\right)}}+C$, where $a=3$, $c=2$ and $n=1$
The integral $\int\frac{1}{36\left(x+3\right)^2}dx$ results in: $\frac{-1}{36\left(x+3\right)}$
$\frac{-1}{36\left(x+3\right)}$
Intermediate steps
Take the constant $\frac{1}{36}$ out of the integral
$\frac{1}{36}\int\frac{1}{\left(x-3\right)^2}dx$
Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(c-1\right)\left(x+a\right)^{\left(c-1\right)}}+C$, where $a=-3$, $c=2$ and $n=1$
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.