Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate by parts
- Integrate by partial fractions
- Integrate by substitution
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
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We can solve the integral $\int\frac{\sqrt{x^2-25}}{x^3}dx$ 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 $\sqrt{x^2-25}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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
Isolate $dx$ in the previous equation
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Rewrite the fraction $\frac{u^2}{\left(u^{2}+25\right)^{2}}$ in $2$ simpler fractions using partial fraction decomposition
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 $\left(u^{2}+25\right)^{2}$
Multiplying polynomials
Simplifying
Assigning values to $u$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{u^2}{\left(u^{2}+25\right)^{2}}$ in decomposed fraction equals
Expand the integral $\int\left(\frac{1}{u^{2}+25}+\frac{-25}{\left(u^{2}+25\right)^{2}}\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int\frac{1}{u^{2}+25}du$ results in: $\frac{1}{5}\arctan\left(\frac{\sqrt{x^2-25}}{5}\right)$
The integral $\int\frac{-25}{\left(u^{2}+25\right)^{2}}du$ results in: $-\frac{1}{5}\left(\frac{1}{2}\arctan\left(\frac{\sqrt{x^2-25}}{5}\right)+\frac{5\sqrt{x^2-25}}{2\left(2^{\left(x^2-25\right)}+25\right)}\right)$
Gather the results of all integrals
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