Final answer to the problem
Step-by-step Solution
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- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
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We can solve the integral $\int\frac{\sqrt{x^2-25}}{x^3}dx$ by applying integration method of trigonometric substitution using the substitution
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
Substituting in the original integral, we get
Simplifying
Factor the polynomial $25\sec\left(\theta \right)^2-25$ by it's greatest common factor (GCF): $25$
The power of a product is equal to the product of it's factors raised to the same power
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
Taking the constant ($5$) out of the integral
Simplify $\sqrt{\tan\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
Apply the trigonometric identity: $\frac{\tan\left(\theta \right)^n}{\sec\left(\theta \right)^n}$$=\sin\left(\theta \right)^n$, where $x=\theta $ and $n=2$
Take the constant $\frac{1}{25}$ out of the integral
Simplify the expression inside the integral
Apply the formula: $\int\sin\left(\theta \right)^2dx$$=\frac{\theta }{2}-\frac{1}{4}\sin\left(2\theta \right)+C$, where $x=\theta $
Express the variable $\theta$ in terms of the original variable $x$
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$
Express the variable $\theta$ in terms of the original variable $x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Expand and simplify