We could not solve this problem by using the method: Integrate by parts
1
We can solve the integral $\int\frac{1}{\left(x^2-9\right)^2}dx$ by applying integration method of trigonometric substitution using the substitution
$x=3\sec\left(\theta \right)$
Intermediate steps
2
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Simplify $\left(\tan\left(\theta \right)^2\right)^2$ 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 $2$
Expand the fraction $\frac{1-\sin\left(\theta \right)^2}{\sin\left(\theta \right)^{3}}$ into $2$ simpler fractions with common denominator $\sin\left(\theta \right)^{3}$
The integral $\frac{1}{27}\int\csc\left(\theta \right)^{3}d\theta$ results in: $\frac{-x}{18\left(x^2-9\right)}-\frac{1}{54}\ln\left(\frac{x}{\sqrt{x^2-9}}+\frac{3}{\sqrt{x^2-9}}\right)$
The integral $\frac{1}{27}\int-\csc\left(\theta \right)d\theta$ results in: $\frac{1}{27}\ln\left(\frac{x}{\sqrt{x^2-9}}+\frac{3}{\sqrt{x^2-9}}\right)$
Combining like terms $-\frac{1}{54}\ln\left(\frac{x}{\sqrt{x^2-9}}+\frac{3}{\sqrt{x^2-9}}\right)$ and $\frac{1}{27}\ln\left(\frac{x}{\sqrt{x^2-9}}+\frac{3}{\sqrt{x^2-9}}\right)$
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
$L.C.M.=\sqrt{x^2-9}$
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Combine and simplify all terms in the same fraction with common denominator $\sqrt{x^2-9}$
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