# $\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$

If $f(x)$ is a continuous function on $\mathbb{R}$ and I am asked to find $\int f(x) \, dx$, what is the problem with the following answer:

$$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$$

There is the occasional situation where this is a useful thing to do.

You are probably not in one of those situations. You are likely in a situation where your goal is something like

Demonstrate understanding of the integration techniques you've learned in this course

• This is the correct answer. The other one is clever, but doesn't get at the heart of the matter. – MathematicsStudent1122 Dec 25 '17 at 6:18

• Answer like this or don't answer at all!! – Shashi Dec 24 '17 at 11:01
• Best answer I saw in a long time – ℋolo Dec 24 '17 at 11:02
• This is funny, but the notation $\int f(x) \, dx$ is asking for an antiderivative of $f$, and it's not tautological that $\int_0^x f(t) \, dt + C$ is an antiderivative of $f$. – littleO Dec 25 '17 at 6:10
• Yes, @littleO, you're right. That fact is the content of the Fundamental Theorem of Calculus, which was referenced in the original title of the post: "What if I answer an integral question using the Fundamental Theorem of Calculus?". – JonathanZ supports MonicaC Dec 25 '17 at 6:29
• @littleO If you so desire, you can read my answer assuming that the first "problem" is "the material problem of the answer" and that the second "problem" is "the problem ascribed to the answer by another person". The reason why any person should have a problem with that answer becomes apparent when one finds himself on the receiving end of a deliberately unsatisfactory answer (as per my second sentence). – user228113 Dec 25 '17 at 11:18

I can't beat @G Sassatelli's answer, but consider what would have happened in your high school class if you were given the problem $$\text{Solve } x^2 -5x + 6 = 0$$ and you had answered $$\{x \mid x^2 -5x + 6 = 0\}.$$

If I were your teacher I'd think "This student really understands set builder notation, but hasn't demonstrated any knowledge of factoring quadratics. Too bad the test was on factoring."

But it is a cute answer that displays a good understanding of the notation, as per @John Coleman. If I were you and considering using it on a test, I would give the definite integral answer first and continue with "while this is 100% valid we can also be more explicit ..." and give the usual answer. That way you've shared your fun answer with the teacher but also demonstrated that you know the methods it was intended to test.

Despite teaching calculus for 20+ years, I have never seen a student do this on an exam, so the novelty of seeing it for the first time would count for something. The first time I would see it, I would accept it and even be somewhat impressed with the student. Far too few students really understand that part of the fundamental theorem of calculus. The second time I would see it, I would suspect that a thoroughly unimpressive student heard about how to get credit for no work. Their plan would backfire and they would get no credit for the problem.

There are cases when it's appropriate do this:

Someone says

$e^{-x^2}$ is not integrable

So we say

Of course it is integrable. All continuous functions are integrable. The integral is $\int_0^x e^{-t^2}\;dt$. What you mean is: the integral is not an elementary function.