- Prove the following identity:
- Using part (a) deduce the formula
- Proof. Following the hint, we make the substitution
So we then have
Here, we change the the name of the variable of integration from to . (We can always rename the variable of integration since integrating is the same as integrating , for example.) So this means we have
- Now to deduce the requested formula, first we use the trig identity , which implies , to rewrite the integral
Then, using the formula we established in part (a) we have
Now, we use the substitution method, letting
So we obtain,
The last equality follows since so that this is an even function. Hence (by a previous exercise) the integral from -1 to 1 is twice the integral from 0 to