Evaluate . Deduce that π does not equal 22/7.
I thought about expanding the numerator and then performing long division, but this way got really messy for me, so I am wondering what the correct method would be.
Evaluate . Deduce that π does not equal 22/7.
I thought about expanding the numerator and then performing long division, but this way got really messy for me, so I am wondering what the correct method would be.
There is an elegant solution using Vandermonde's substitution.
And it isn't really messy if you properly expand :
.
Now,
And since (see here), we have
Thus .
Since for , we conclude that
(In fact, , since supposing otherwise would imply that the integral is negative).