2nd Fundamental Theorem of Calculus

Warning. Most of people's biggest motivation is grade. Let me tell you what you are NOT supposed to do on the exam: $$\int_{1}^{x}\cos(x)dx.$$ What's the trouble? First, since the integral has bounds, it is a definite integral. In the way that we have treated integral, we need to treat $x$ like an arbitrary fixed real number. However, that makes no sense for $dx$ to certain mathematicians, especially geometors, who always writes $$df(t) = f'(t) dt,$$ so if $x$ is constant, then $dx$ would be zero to them, making the whole integral zero.

Remark. You don't really need to understand what I said about why geometors might hate writing $x$ on the bound and then use it for integrating variable (a.k.a. "dummy variable"). What's important is that you must NOT write such expression because many graders like geometry!

What's better? Use two different letters, for instance, we may write $$\int_{1}^{x} \cos(t)dt.$$ Now, note that we can define a function $F(x)$ in $x$ by saying $$F(x) = \int_{1}^{x}\cos(t)dt.$$ Is this legit? If you plug in a real number in $x,$ the right-hand side will give you exactly one real number. Hence, yes, it is legit. More explicitly, we have $$F(x) = \int_{1}^{x}\cos(t)dt = -\sin(x) + \sin(1),$$ applying 1st FTC. In particular, we have $F'(x) = \cos(x).$

Exercise. Let $$F(x) = \int_{0}^{x} \frac{1}{\cos^{2}(t)}dt$$ with $-\pi/2< x < \pi/2.$ What is $F'(x)$?

Exercise. Let $$F(x) = \int_{0}^{x} t^{2} + 5t^{4}dt.$$ What is $F'(x)$?

Exercise. Let $$F(x) = \int_{-x}^{x} \frac{1}{e^{t}}dt.$$  What is $F'(x)$?

Q. Now, suppose that $$F(x) = \int_{0}^{x} \cos(t^{2} + \cos(t))dt.$$ How would you compute $F'(x)$?

Well, if you have gone through the above exercises you might guess that the answer is: $$F'(x) = \cos(x^{2} + \cos(x)).$$ Unfortunately,the methods that you used for the above exercises do NOT apply to give you this conclusion.

What can we do? This is where you need to meticulously dig into the details of definitions of a derivative and an integral. (If you enjoy this process, you might consider majoring/minoring in mathematics.) Luckily p. 336 and 337 of your book does this, so if you are interested read this. Anyways, our guess is correct:

2nd Fundamental Theorem of Calculus (cf. Theorem 6.2 on p.336). For any continous function $f(x)$, we have $$\frac{d}{dx}\int_{a}^{x}f(t)dt = f(x).$$ Now, we can do the following exercise.

Exercise. Let $$F(x) = \int_{0}^{x} \cos(t^{2} + \cos(t))dt.$$ What is $F'(x)$? (Hint: Use 2nd FTC.)

Exercise. Let $$F(x) = \int_{0}^{x^{2}} \cos(t^{2})dt.$$ What is $F'(x)$? (Hint: Use 2nd FTC.)

Exercise. Compute $$\frac{d}{dx}\int_{0}^{\cos(x)} 3000^{t^{2}}dt.$$

Exercise. Compute $$\frac{d}{dx}\int_{-x}^{\cos(x)} 3000^{t^{3}}dt.$$