Final answer:
The integral ∫(dx)/(x²√(4 - x²)) can be evaluated using trigonometric substitution by setting x = 2sin(θ), resulting in an expression involving the inverse sine function.
Step-by-step explanation:
To evaluate the integral ∫(dx)/(x²√(4 - x²)), we can make use of trigonometric substitution. Specifically, we can let x = 2sin(θ) because this makes the term under the radical become 4 - 4sin²(θ), which is a perfect square related to the Pythagorean identity sin²(θ) + cos²(θ) = 1. Thus, √(4 - x²) becomes 2cos(θ).
After substitution, we will need to express dx in terms of dθ. With x = 2sin(θ), we find that dx = 2cos(θ)dθ. Plugging into the integral gives us ∫1/(4sin²(θ) * 2cos(θ)) * 2cos(θ)dθ, which simplifies to ∫(1/4sin²(θ))dθ. This integral can now be computed using a trigonometric identity.
You might also recognize that this is the derivative of a standard inverse trigonometric function, which can further simplify solving the integral. The solution involves finding a trigonometric expression for θ and then substituting back to get the answer in terms of x, which will likely involve using the inverse sine function, or arcsin(x/2).