Final answer:
To evaluate the indefinite integral ∫√(x²-9)/(x⁴) dx, we can split it into two parts: one for the numerator and one for the denominator. You can then make appropriate substitutions to simplify the integrals.
Step-by-step explanation:
To evaluate the indefinite integral ∫√(x²-9)/(x⁴) dx, we can start by simplifying the expression under the square root. Note that x²-9 can be factored as (x-3)(x+3). So, the integral becomes ∫√[(x-3)(x+3)]/(x⁴) dx. We can now split the integral into two parts: one for the numerator and one for the denominator.
Let's start with evaluating the integral ∫√(x-3)/(x⁴) dx. To do this, we can make a substitution u = x-3, which gives us du = dx. So, the integral becomes ∫√u/(u+3)⁴ du.
Next, let's evaluate the integral ∫√(x+3)/(x⁴) dx. We can make a substitution v = x+3, which gives us dv = dx. The integral then becomes ∫√v/(v-3)⁴ dv.