Evaluate # int cos(x^2) dx # using a power series?

1 Answer

# int \ cos(x^2) \ dx = c+x - (x^5)/(5 * 2!) + (x^9)/(9*4!) - x^13/(13*6!) + ...#

Explanation:

Starting with the Maclaurin Series for #cosx#

# cosx = 1 - x^2/(2!) + x^4/(4!) - x^6/(6!) + ... \ \ \ \ \ \ AA x in RR#

So then:

# cos(x^2) = 1 - (x^2)^2/(2!) + (x^2)^4/(4!) - (x^2)^6/(6!) + ... #
# " " = 1 - (x^4)/(2!) + (x^8)/(4!) - (x^12)/(6!) + ... #

Then we have:

# int \ cos(x^2) \ dx = int \ 1 - (x^4)/(2!) + (x^8)/(4!) - (x^12)/(6!) + ... \ dx#
# " " = x - (x^5)/(5*2!) + (x^9)/(9*4!) - x^13/(13*6!) + ... +c#