Antiderivative of sqrt (1+cos^2x)?

Question

How can we find it?
thanks in advance

Answer 1

There is no elementary antiderivative for
this function. Actually, this integral can be
reduced to a standard elliptic integral of the
second kind. Here's how it's done:
∫ √(1+cos² x) dx = ∫ √(1 + 1-sin² x) dx
= ∫ √(2- sin² x) dx = √2 *∫ √(`1-½ sin² x)dx
= √2 E(x, √2/2).
This integral arises in finding the arclength
of the sine curve.
For example, suppose we want to
find the arclength of the sine curve from 0 to π/2.
Then we have L = √2*E(√2/2), where
E is now the complete elliptic integral of the second kind.
Looking this up in a table, we get L = √2 *1.30564 = 1.9101
(approximately).