Evaluate the integral by using multiple substitutions.

To evaluate the integral given, we start by identifying the substitution that helps simplify the integral effectively.

First, let’s recognize what substitution can help us reduce complexity. If we let , then . Rewriting our integral using this substitution gives us a simpler form to manage:

Next, another useful substitution can be which helps because the derivative implies . This leads us to convert the integral to a form involving .

After integrating with respect to , we can backtrack to our original variable by substituting back into the equation.

However, it’s wise to remember that common mistakes here could involve improper limits or failing to simplify adequately into a usable form. So keep a check on your algebraic transformations and use basic trigonometric identities whenever necessary!

By applying these strategies effectively, you can navigate through the integral and come up with the solution much more smoothly. Have fun with those substitutions!