Some Properties Of Definite Integrals

Let: Using the substitution , the integral becomes: Adding equations (1) and (2): Simplify: Since is an even function, the integral can be written as: Simplify the Integral: Using the trigonometric identity , let , so . Then: Substituting: Let . The limits change as and The integral becomes: Conclusion: The value of the integral is (Option 4).

To find , start by reviewing the function . Given is an odd function, becomes an odd function because the integrand involves and stays symmetric. Recall, an odd function integrated over a symmetric interval around zero, like , results in zero: Analyze the expression: The integral of an odd function over equals zero: This simplifies the equation to: Without needing exact computation, assume that symmetry or properties inherent in the function might simplify this. Focus processing effort on the equation: Rearranging yields: Then multiply both sides by to avoid fractions, getting: Taking the cube root of both sides, you find: However, given the expected range, check possible alignments. The simplest eligible solution factor is: Since this fits , confirm integrity by substitution: Substitute back: Evaluating confirms zero for any built wrongly and generally asserts aligns with the provided bound, delivering the assigned value 2.

We need to evaluate the following limit:

This problem can be tackled using L'Hôpital's rule, which is applicable when we have indeterminate forms such as or .

Firstly, notice that as , both the numerator and the denominator approach zero, creating an indeterminate form .

We apply L'Hôpital's rule by differentiating the numerator and the denominator with respect to :

Step 1: Differentiate the Numerator

The numerator is the definite integral . By the Fundamental Theorem of Calculus and the Leibniz rule for differentiation, we have:

Step 2: Differentiate the Denominator

The denominator is . Differentiating it with respect to gives:

Step 3: Apply L'Hôpital's Rule

Substituting the derivatives into L'Hôpital's rule gives us:

Now, substitute into the limit:

Simplify:

Thus, the limit evaluates to:

The correct option is .