Definite Integral

Answer (b)

Given,







On comparing,
and

Now,

Step 1: Recognizing the summation Rewriting, resembles the Taylor series of , leading to: Step 2: Evaluating the integral

Step 1: Split the integral based on the absolute value We are asked to evaluate: First, break the absolute values into pieces based on the points where the expressions inside the absolute values change sign: - changes sign at , - changes sign at . Thus, we split the integral into two parts: Step 2: Evaluate the integrals Now evaluate each part: For : For : Thus, the value of the integral is:

Step 1: Substituting using symmetry Let: Using the property: Applying transformations and simplifications, we get: Step 2: Solving for

Step 1: Splitting the integral Rewriting: Splitting: Step 2: Using substitution Let , then . This simplifies the integral, leading to:

Step 1: Substituting in terms of tan Rewriting in terms of : Using , we get: Step 2: Using substitution Let , then . The integral simplifies to: Step 3: Evaluating the integral

Step 1: Simplifying the integrand The given integral is: Notice that the integrand involves symmetric limits of integration, i.e., from to . Let's check if the two terms and are symmetric. Since the integrand is an odd function (the subtraction of two functions that are symmetric about the origin), the integral of an odd function over a symmetric interval from to is zero. Thus, the value of the integral is:

Step 1: Expanding the function Expanding : Rewriting the integral: Step 2: Splitting into separate integrals Using integration by parts, where , : Applying integration by parts repeatedly: Similarly, Step 3: Evaluating and summing terms

Step 1: Substitution Let: Then , and when , , and when , . The integral becomes: We can break the integral into two parts: Step 2: Evaluate the first integral Step 3: Evaluate the second integral Step 4: Combine the results Thus, the value of the integral is: