Question

Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.)

∫ (x²-4)/xdx, x = 2 sec(θ)

Answer 1

To evaluate the integral ∫ (x²-4)/x dx with the substitution x = 2 sec(θ), replace x and compute the resulting trigonometric integral. Simplify and integrate, and then substitute back to x, adding the constant integration, C.

Step-by-step explanation:

To evaluate the integral ∫ (x²-4)/x dx using the trigonometric substitution x = 2 sec(θ), first substitute x with 2 sec(θ). This substitution simplifies the integral by using trigonometric identities. The differential dx will be (2 sec(θ) tan(θ) dθ) deriving from the derivative of x with respect to θ.

Now, express the integral in terms of θ:

∫ ((2 sec(θ))² - 4) / (2 sec(θ)) (2 sec(θ) tan(θ) dθ)

Simplify the integral and evaluate it, followed by back substitution to return to the variable x. After computing the integral, you would add the constant of integration, C, to represent the indefinite integral.

It's essential to check for completeness in learning how to perform these types of integrations, particularly ensuring that all steps are correctly followed, from the trigonometric substitution to the simplifications and proper back substitution.