I want to solve this integral quarrel!

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I'm at odds with my calculus II teacher because of this integral: -4/(1+X^2) DX.

My teacher says it's -4 arc tan X + C while I say it is 4 arc cotan X + C.

Anyone can help me?

-4 arc tan X + C. Maybe yours is a rearranged version of the same? That happens occasionally when taking integrals. How did you arrive at your answer?

...and show your work...

I think your teacher is right in this case

-4arctan(x) + C is correct.

Where the heck did you even get an arccotan from? I've never had to work with one of those in all my years as an undergrad (although my school is known to be pretty slackerish...)

let arccotan(x) = y
x = cot(y) : want dy/dx
x = 1/tan(y)
tan(y) = 1/x
(sec^2(y))dy/dx = -1/x^2
(1+tan^2(y))dy/dx = -1/x^2
(1 + 1/cot^2(y))dy/dx = -1/x^2
But y = arccot(x) => cot^2(y) = x^2
=>
(1+1/x^2)dy/dx = -1/x^2
((x^2+1)/x^2)dy/dx = -1/x^2
dy/dx = -1/(1 + x^2)

Therefore, d/dx 4arcot(x) = -4/(1+x^2)

Most impressive young one. You shall soon be a jedi.

(someone check my math. I'm sleep deprived and stressed out...)

It's -4 arctan x + C.

It's highly unlikely that 4 arccot x + C is another answer, since it sort of implies that arctan x = - arccot x + C for some constant C, which is almost certainly false.

twoshots - your derivation looks right though I didn't check it in detail, but it seems like the proper way to solve it.

I've never had to deal with arccot either, but I can imagine that's just because I didn't take calc 2 at my university (I transferred it in) and I hear that the calc 2 here is so insanely hard, it deals with a ridiculous amount of calculus with trig, I can certainly imagine an arccot popping up somewhere in there.

You are, of course, both right.

The angle whose tangent is "x", is pi/2 - (the angle whose cotangent is "x").

Just think of a right-angled triangle.

I.e arc cotan X = pi/2 - arc tan X.

Hence the two solutions you give above do indeed just say exactly the same thing, bar for the choice of constant "C".

Now, as to which one of you is "more right"... I'd side with your teacher, to a tiny degree. Although they have to have a minus sign in their version of the solution, in general usage, the arc tangent is the "preferred" function. It mainly a matter of taste.

Last edited by lau on 01 May 2008, 2:06 pm, edited 1 time in total.

Good choice of words - "almost certainly false".

Indeed, we do have exactly C = (2n+1/2)pi as the choices of constant.

It's because my teacher did it this way: -4*1/(1+X^2) = -4 arctan X + C

And I did it that way: 4*-1/(1+X^2) = 4 arccot X + C

Here we go for each other's work

shoot, you're right. teaches me to skim math

I thought that was sterile squabbling about indefinite integration...

I prefer definite integrals better, both single and multi-variable

Ah! So - you're not a mathematician, then. A physicist, maybe, or an engineer? Surely not a statistician!

Remind me to never take a calc course. This stuff gives me headaches.

But, depending on C, they ought both be right... or at least feasible.

Though I could be wrong. Last time I did calc was for fun... that was middle school, back when I still found maths fun.

*takes asprin*

You bet!

I learn calculus for its applications; I know someone else who prefer vectorial spaces and indefinite integrals. As for me, definite integrals are the prettiest equations within my mathematical knowledge.

And often can be subdued and approximated with a numerical method

The way to see which answer is right is to take the derivative of both answers and see which one will give you the expression under the integral. If you do that, you will easily see your teacher is right.

The formula to memorize is that integral of dx/(x^2+1) is arctangent. I think you just memorized cotangent instead of arctangent because the words are similar. So try to memorize again that it is arctangent and not cotangent.

Again you can check it by taking the derivatives.

Also there is no minus sign either. There are other integrals with a minus sign though. Maybe you confused this one with those other ones. Again, just try to memorize where you have minus sign and where you don't. *AND* as you do it keep checking it with the derivatives.