as presented by William Dunham in his book Euler, The Master of Us All. First post here.
The first step is to recall a standard trigonometric substitution in calculus:
We're interested in the integral:
Substituting with x we see that:
And the integral is
Now Euler makes a complex change of variable:
The last step is another standard result from calculus which I will assume for the time being (more here).
Undo the substitution:
We will use two identities involving i:
(For the second one, see the previous post). Now: