There has to be a better way to compute integrals than looking at paper with vacant eyes and waiting for inspiration or burning a hole through paper with trial-and-error. Share your secrets with a brainlet.

# There has to be a better way to compute integrals than looking at paper with...

Nope, integration is harder than differentiating. IIRC, this was actually shown mathematically (meta stuff, the likes of Gödel).

Way back in Freshman Calc, I had a prof who said that while any old fool can differentiate, 'Integration is an art'

Like with everything, I guess, just practice. In physics, especially, you'll find the same integrals coming up again and again, so eventually you'll just be able to knock them out without even thinking about it

the problem is, that there are loads of stuff that cannot be integrated properly and instead you get non-trivial functions like erg(x) or Si(x) or maybe the occasional arctan(x).

I have HEARD that there's an algorithm which can perform general integration. I mean, analytically, not numerically.

They tested it on a big book of integrals and found a huge number of them were wrong. Mostly typos. A plus where a minus should be. That sort of thing.

Never came across it again. Most people (except physicists and mathematicians) just brute-force it with a computer nowadays.

Try Wolfram.

You mean the Risch algorithm.

It's implemented in Wolfram alpha

The specification is about 100 pages long

Are there any integrals that aren't tabulated or given methods/algorithms to solve them?

example, integral table off google.

integration by parts

arctan, arcsin, etc.

some other shit I fucked up so bad I forgot what it even is already

absolute bane of my existence in high school calc and probably will continue into college. I assume practice is really the best way to fix this, right?

There is a homotopy type of method that introduces an extra parameter and then utilizes differentiation under an integral.

Feynman used it to calculate definite integrals that other people struggled with.

**en.wikipedia.org**

yep, just lots of practice.

But calc practice is easy cuz the problems are usually solvable in half a page.

Thanks. I never would have located that on my own. I'll probably never use it, but it's nice to know it exists.

's prof plagiarized that bon mot. :)

We were taught at calculus classes that integration is an art, not a science (in contrast to differentiation—even a monkey can be trained to take derivatives). And we were taught wrong. The Risch algorithm (which is known for decades) allows one to find, in a finite number of steps, if a given indefinite integral can be taken in elementary functions, and if so, to calculate it. **link.springer.com**

There has to be a better way to compute integrals than looking at paper with vacant eyes and waiting for inspiration or burning a hole through paper with trial-and-error.

Yes, it is called a computer, specifically the Risch algorithm.

If you don't mind me asking, where are you from that you do integration in high school calc? I took AP calculus in high school and the farthest we got was differentiation. The only time we did integration was when the instructors were showing us what we would do when we get to college. The integration we did in high school was entirely extra credit.

california bay area, doing calc BC. AB had only a little integration, and the majority of kids don't even take calc at all, let alone BC. in BC we've done a lot of integration, but this is a class which students chose specifically to take.

that's what I figured, although I'm a brainlet so I like to use an entire page for some problems just so I can be very careful with it

But integration by parts as only a little bit? I didn't do that until Calculus II in college. Hell we didn't even do vector calculus until Calculus IV.

Actually when I talked to my Canadian friends about Calculus IV, they had no idea what I was talking about. Is this an American thing? Are Canadians just retarded? Am I retarded? Who is retarded?

shit wasn't good

what exactly do you mean?

well we did quite a bit of integration by parts this year, but I can't really say how in-depth we actually went. In fact we just finished up with vector calculus (pretty basic stuff, dot products, cross products and the like, practically not even calculus really) and that's it for this year. we're done with the curriculum and looking towards reviewing everything before college and for finals and AP testing.

I'm pretty sure I'm retarded since I struggle to do work properly in all of my classes but who knows, you could be retarded too.

In fact we just finished up with vector calculus (pretty basic stuff, dot products, cross products and the like, practically not even calculus really)

What you are describing is algebra.

Definition: the set of classical analytic functions is defined as [math]\sin[/math], [math]\cos[/math], [math]\tan[/math], [math]\csc[/math], [math]\sec[/math], [math]\cot[/math], [math]\arcsin[/math], [math]\arccos[/math], [math]\arctan[/math], [math]\exp[/math], [math]\ln[/math], [math]\left|\cdot\right|[/math], [math]\sqrt[n]\cdot[/math] for [math]n\,\in\,\mathbf Z[/math] constant, polynomials, and their linear combinations, products, reciprocals and compositions.

Theorem 1: all classical analytic functions are computably differentiable and their derivatives are classical analytic functions.

Proof: Cf. your calc 1 & 2 lessons.

Theorem 2: there exists a classical analytic function whose counter-derivative isn't a classical analytic function.

Proof: [math]\begin{array}[t]{cccl} f: & \mathbf R & \longrightarrow & \mathbf R \\ & x & \longmapsto & \exp\left(-x^2\right) \end{array}[/math].

Corollary: integrating classical analytic functions is more difficult than differentiating them.

Dude, take the obvious derivatives as tangent lines and then just do the opposite. In fact, if you do that to an iterative depth on a function, you'll get a shape with sharp edges. Compare x^2 with x^6. Then you can find Area with geometry

Integration by parts is taught at highschool in Europe. Then again, more rigerously in a first year course, which goes all the way up to vector calculus.

i.e Your country is retarded.

like with everything I guess, just practice

Or use a calculator, that's much more efficient. Honestly, the mathematician's obsession with solving problems by hand is something I doubt I'll ever understand.

My mother never let me use a calculator as a kid. On standardized tests I was given one and the only operators I used were the elementary ones (only + and • before I learned how to manipulate - and ÷--around 6th or 7th grade iirc). I hated it but I was able to pick up on patterns easily compared to my peers who had used a calculator throughout their academic career.

For me, it was mainly a matter of

I'm tired of doing so many steps. Is there a faster way?

this

reminder that CAS and automated provers are becoming better mathematicians than mathematicians were ever capable of being

yeah well that was our final mini- "unit" for this year, I know it's just algebra. although we tied it back in to calc with some derivatives, but that was about it for vectors this year.