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
$u=\sqrt{x^2-25}$
Intermediate steps
2
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
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 $\left(u^{2}+25\right)^{2}$
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 $\int\frac{-25}{\left(u^{2}+25\right)^{2}}du$ results in: $-\frac{1}{5}\left(\frac{1}{2}\arctan\left(\frac{\sqrt{x^2-25}}{5}\right)+\frac{5\sqrt{x^2-25}}{2x^2}\right)$
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.