What is the exact value of 4 times the integral from -1 to 1 of dx/(1 - x^2)?

To determine the exact value of 4 times the integral from -1 to 1 of the function ( \frac{1}{1 - x^2} ), we first need to evaluate the integral itself.

The integral can be expressed as:

[

\int_{-1}^{1} \frac{dx}{1 - x^2}

]

This integrand can be recognized as having singularities (points where it approaches infinity) at ( x = -1 ) and ( x = 1 ). Thus, we would typically evaluate this integral using improper integrals, breaking it into two parts:

[

\int_{-1}^{1} \frac{dx}{1 - x^2} = \lim_{\epsilon \to 0} \left( \int_{-1}^{-\epsilon} \frac{dx}{1 - x^2} + \int_{\epsilon}^{1} \frac{dx}{1 - x^2} \right)

]

The function ( \frac{1}{1 - x^2} ) can also be rewritten using partial fractions:

[

\frac{1}{1 - x^2} = \frac{1}{(1 -