Integrate by parts:
integrate tan^-1 y dy
i just can’t do it, guys i suck at math…
Well, I think you’re going to need substitution as well, after integrating by parts. I dunno if you learned this using f(x) and g(x) as the equations, or what. We used u and v, so if that’s not how you learned, just plop your notation of choice in there instead.
(integral) arctan y dy = (integral) u dv
u = arctan y
du = dy/(1+(y^2))
dv = dy
v = y
So,
(integral) u dv =
uv - (integral)v du =
(y(arctan y) - (integral)(y/(1+(y^2))dy) + C
Obviously, you still need to figure out (integral)(y/(1+(y^2))dy, which is where substitution comes in. I don’t want to do the whole problem for you, so let me know if you still need help.
I remember this same problem that stumped me back in freshman calc. I don’t recall the exact process, but I do remember that tan[sup]-1[/sup]y can be re-written into a form that is much easier to integrate.
Thanks
and yeah we use u and v
Isn’t 1/tan y the same as cot y (cos y/sin y)?
It isn’t clear whether tan^-1 represents 1/tan or arctan. If it’s 1/tan, you can let x = sin y; dx = cos y dy.
If it’s arctan, well it’s been too long since I studied calculus to do that off the top of my head.
It isn’t 100% absolutely, totally, crystal clear, but the vast majority of the time, it means arctan. In fact, I have never assumed that it meant arctan and gotten it wrong.
Just as a helpful hint, try integrals.com. For this one, enter the following (capitalization and bracketing matter):
ArcTan
Imagine that, in this day of Maple and Mathematica they are still turning out calc students who can (or can’t) integrate arc tan and still don’t understand the first thing of what calculus is about! Find me the student who can explain clearly why the derivative of the integral of a continuous function is the original function. Not prove it (anyone could memorize a proof), but explain it–it ought to be obvious.
Oh, Hari, now you made me feel stupid! I took two calc classes, about 3 or four years ago, and I can answer neither of these questions. But, on the other hand, isn’t it amazing some students at university have to take biology and still not know or be able to explain evolution completely? I mean, shouldn’t it be obvious? :rolleyes:
Since the OP specified integration by parts, I assumed “tan^-1” meant arctan, since, as you said, integrating coty dy just involves a trivial substitution.
I reject that analogy. The equivalent for evolution is that during reproduction, the copying isn’t perfect and among the mutations some give, by pure chance, advanatages and these are selected for. Now was that so hard.
I have no intention of hijacking this thread. However,
Your explanation wasn’t quite complete. You didn’t explain evolution, you just summarized it (in a very vague way that doesn’t come close to explaining the whole theory - or theories, as some see it), kind of like how “integration by parts” rules summarize how to solve the above problem, but not explain the meaning of the work. True, it isn’t the best analogy, since math is much more a rules-driven field than evolutionary biology, but my point was just to show that its no big deal if a particular student has trouble with calculus. The fact is, most people take it only because its required, and then never have to derive or integrate a function ever again in their entire lives. So why would they care to learn the reasoning behind “every stupid math rule, just so they can pass the damn course and move on with their lives?”
Perhaps a better analogy would be to ask students with introductory chemistry to explain Shroedingers Equation…its just a silly expectation, and one that most students wouldn’t try to live up to, since its not their field, and they don’t care. Now if Quantum Chemists couldn’t explain Shroedinger’s equation…THEN I’d worry about the school system!!
Yes, well that sort of summary is all I wanted. The derivative of the integral is simply the integral from x to x + h, divided by h, where h is infinitesimal. (More precisely, the ordinary part of that.) In such an interval, the function is essentially constant at
f(x) and the integral is just hf(x) and when you divide by h you get f(x). The point is that if you understand calculus it is all clear and if you don’t it isn’t. This is not a proof but the proof is almost an anticlimax.