Integral Calculus

Step 1: Evaluating
Consider the given integral: We use the transformation , so . Under this transformation: Rewriting as , we obtain: Thus, we conclude: Step 2: Evaluating
Similarly, considering the symmetry properties of the integral: Using the substitution , it can be shown that: Step 3: Conclusion
Thus, we have:

Step 1: Substituting a Trigonometric Identity
Let: Rewriting the integral in terms of : Step 2: Simplifying the Expression
Using the identity: Splitting the integral: Step 3: Evaluating the Integral
After integration and simplification: Step 4: Conclusion
Thus, the final answer is:

Step 1: Letting be the given integral
Using the change of base formula for logarithms: Rewriting the integral: Step 2: Substituting
Let: Rewriting the integral: Since , we multiply by and integrate: Using the integral formula: Applying this: Step 3: Conclusion
Thus, the final answer simplifies to: