Second Fundamental Theorem

The Fresnel Cosine Integral $C(x)$ is defined by

$$C(x) \, \equiv \, \int_0^x \cos( \frac{\pi}{2} t^2) \, dt \;.$$ (1)

This integral arises in certain optics applications.

1. Sketch the general shape of $C(x)$. (You'll probably need to sketch the integrand, $\cos( \frac{\pi}{2} t^2)$, first.)

2. Is $C(x)$ an even or odd function?

3. Evaluate the following:

   
   

   $$\frac{d}{dx} C(x)$$ (2)

   
   

   
   

   $$\frac{d}{dx} 5C(x)$$ (3)

   
   

   
   

   $$\frac{d}{dx} C(x^3)$$ (4)

   
   

   
   

   $$\frac{d}{dx} C(1/x)$$ (5)

   
   

4. Do this only if you have some extra time on your hands. As you might remember from precalculus, the cosine function can be written as an infinite series:

   
   

   $$\cos(x) = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 - \frac{1}{6!} x^6 \ldots \;.$$ (6)

   
   
   (We will derive this formula when we do chapter 9.) Use the above equation to come up with a series expression for the integrand. Then, integrate term-by-term to come up with a series expression for $C(x)$. This series is (I suspect) how Maple calculates $C(x)$ to make pretty plots.