# Differential calculus vs Integral calculus

I have never done integration in my life and I am in first year of university. Is it harder than taking the derivative? I've heard its just going backwards. Also, my high school taught me only differentiation, I don't know why we never touched on integration. I'm going to be starting it next week and I want to know what I'm facing. Is it generally considered harder than differentiation? Thank you in advance.

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It depends on what you want to integrate, in general though it can be much harder than differentiation. There is a saying: "Differentiation is a technique, integration is an art". Some integrals, even though they exist, may not have a closed form. –  Rogelio Molina Oct 12 '13 at 0:48
Yes. There are a lot of functions you can write down easily whose integrals don't exist (as "elementary" functions like you see in calculus clas). Often it is not easy to know how to begin to do an integral. On the other hand, if you know the rules, taking the derivative of just about any function is just a matter of applying the rules. –  Stefan Smith Oct 12 '13 at 0:50
It's often easier to prove that a function is integrable than to prove that one is differentiable, because functions are integrable under much weaker conditions. Actually integrating is another story. –  dfeuer Oct 12 '13 at 0:56
@dfeuer : by "integrable", do you mean that a classical (pointwise) antiderivative exists? –  Stefan Smith Oct 12 '13 at 1:41
@StefanSmith, it doesn't matter, does it? Mathematicians have to study more complicated functions to find ones that are hard to prove integrable. –  dfeuer Oct 12 '13 at 1:46
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## 1 Answer

If you were fine with derivatives, you will be fine with integrals in 1st year calc. It never hurts to pay attention in class (which kind of implies attending class) and to do your homework. In fact, if you have trouble with a problem, you should do more of the same kind as soon as you know the answer. As to difficulty:

Integrals start out harder than derivatives and wind up easier. The reason derivatives are easier is that if a function has a derivative you can compute what it is. There is an algorithm for doing so. Sometimes the computation may be long and complicated. But theoretically anyway, anyone can do this.

With the integral, you will be given a lot of problems to solve, but there is no algorithm. The kind of problems you get in first year calculus will be solvable if you learn enough tricks. (They are chosen to be solvable). There are hundreds of tricks because over the course of many years lots and lots of smart mathematicians have worked them out. You'll probably learn 3 - 5 tricks in your first year class.

Integrals which are not subject to the tricks (that is, most of them) are evaluated with numerical or approximating methods and for practical purposes that works very well. In fact, with the availability of computers, many of the old tricks have fallen into disuse, because it is faster to just do the numerical computation.

Integrals and derivatives are the reverse of one another in the same sense that addition and subtraction are. As you know, taking an operation in one direction is often easier than reversing it. Thus multiplication is easier than division,and raising things to powers is easier than the reverse: finding roots.

One expression of the connection between derivatives and integrals is the Fundamental Theorem of the Calculus, which you will probably be taught. However, the two subjects are more intertwined than the Fundamental Theorem suggests. Indeed, I find the myriad entanglements fascinating.

Now why does the difficulty turn around later? Derivatives are about differences and division which make a function less smooth; and the smoother a function is, the easier it is to work with. For example, if f is differentiable, f' may only be continuous; and in fact it can be discontinuous. You've taken a relatively smooth function and degraded it.

Integrals are about addition, that is to say averaging, which always smooths things out. You can integrate discontinuous functions and they will come out continuous. Integrate again and they will be differentiable, which is even smoother than continuous. We like smooth.

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