Definite Integral

We are integrating: Let’s analyze the function
Note that: -
- So
Now, consider behavior of :
Case 1:
- (since )
- So
Case 2:
-
- So
At :
At :
Now integrate: Total:

This is a limit form of a Riemann sum: Let , then:
-
-
-
From the pattern, the term becomes: So, the function under the integral is:

Step 1: Evaluate the summation.

Step 2: Substitute into the integral.

Step 3: Evaluate .}
Using , we get .
Step 4: Calculate .

Step 5: Conclusion.
The value of the integral is .