Integration

Step 1: Apply Integration by Parts Let: Using the integration by parts formula:
Step 2: Simplify the Integral After performing the necessary integrations and simplifications, we arrive at the final expression for the integral:
Step 3: Conclusion Thus, the integral is .

We are tasked with evaluating the integral:
Step 1: Simplify the integrand
We can use the identity , so the integrand becomes:
Step 2: Use trigonometric identity
Now use the identity , so the integrand simplifies to:
Step 3: Perform the integral
The integral of over the interval from 0 to is known to be .
Step 4: Conclusion
Thus, the value of the integral is , corresponding to option (c).

Step 1: Recognizing the Integral Form
The given integral is: This is a standard integral, and the solution can be derived using the substitution method.
Step 2: Apply Integration Formula
The integral has a standard solution:
Step 3: Conclusion
Thus, the solution to the integral is .