What is the integral of #cos(t^2)#?

Question

Isaiah Allen · Accepted Answer

#int_(-1)^(1) cos(t^2) dt = 1.8090#

Well, the only way I know how to do this without using functions I don't know (or looking up Fresnel integrals) is to expand this into a Maclaurin series to estimate an easier form of the function to integrate. If you want something centered at a different #x = a#, you would replace this method with a Taylor series...

#cos(t^2) = |[sum_(n=0)^(oo) ((-1)^(n) x^(2n))/((2n)!)]|_(x = t^2)#

#= sum_(n=0)^(oo) ((-1)^(n) t^(4n))/((2n)!)#

#= 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320) - . . . #

graph{(y - cos(x^2))(y - 1 + x^4/2 - x^8/24 + x^12/720 - x^(10)/(40320)) = 0 [-4.934, 4.935, -2.464, 2.47]}

Here, the bottom graph is the estimate graph from the Maclaurin series.

If we truncate this at the fifth nonzero term, we can at least get the integral near #t = 0#:

#int cos(t^2) dt ~~ int 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320)dt#

#= t - t^5/10 + t^9/(216) - t^(13)/(9360) + t^(11)/(443520)#

for #t# really close to #0#.

To compare, I used Wolfram Alpha to get:

#int_(-1)^(1) cos(t^2) dt = ul(1.8090)color(red)(4848 cdots)#

#int_(-1)^(1) 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320)dt = ul(1.8090)color(red)(5009cdots)#

The only drawback is that this approximation only works near #t = 0#.

#int_(-1.5)^(1.5) cos(t^2) dt = ul(1.79)color(red)(837cdots)#

#int_(-1.5)^(1.5) 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320)dt = ul(1.79)color(red)(601cdots)#

#int_(-2)^(2) cos(t^2) dt = color(red)(0.92292292cdots)#

#int_(-2)^(2) 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320)dt = color(red)(0.59954860cdots)#

#int_(-2.5)^(2.5) cos(t^2) dt = color(red)(1.21062cdots)#

#int_(-2.5)^(2.5) 1 - t^4/2 + t^8/24 - t^12/720 + t^(10)/(40320)dt = color(red)(-10.9426cdots)#

Cameron Alexander · Answer

There is no elementary solution.

We want to find: # int \ cos(t^2) \ dt # As indicated in the alternative answers we can find a Maclaurin Series expansion, Unfortunately there is no elementary solution to the integral. Instead, Numerical techniques are used to evaluate the following definite integrals: # S(x) = int_0^x \ sin(t^2) \ dt #, and # C(x) = int_0^x \ cos(t^2) \ dt # Which are known as the Fresnel integrals. Typically the values of these function are calculated using computer algorithms, or looked up in tables (in a similar way to that of the Normal Distribution ). Note: If tables are used, then sometimes the integral functions are normalised with a factor #sqrt(2/pi)# (again in a similar way to the Normal Distribution normalising factor #1/sqrt(2pi)#).

Paisley Johnson · Answer