Question

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{2 x}{\sqrt{x^{4}-9}} d x, \text { Formula } 25 $$

Short answer

The indefinite integral of \(\int \frac{2 x}{\sqrt{x^{4}-9}} d x\) using formula 25 is \(\ln |x^2 + \sqrt{x^4 - 9}| + C\).

Step-by-step solution

Identify the Form

Look for the general form in our integral which matches the formula 25 ordinarily presented in integral calculus, which usually is the integral in the form \(\int \frac{u'}{\sqrt{u^2-a^2}} du\). Here, we can identify \(u = x^2\), \(u' = 2x\), and \(a^2 = 9\). So our integral is indeed in this form.

Apply the Formula

The formula for the integral in this form is \(\ln |u + \sqrt{u^2 - a^2}| + C\). Substituting our values we have the solution as \(ln |x^2 + \sqrt{(x^2)^2 - 9}| + C\). Simplify the expression inside the square root.

Final Step - Simplify

This results in \(\ln |x^2 + \sqrt{x^4 - 9}| + C\) as the final desired integral.