A calculus question! Or three!

Loren Michael

Hey, so, I'm in a situation where I'd like to know some answers but no one around me is in a position to give me any. I'm not a math whiz, which has led me to the following quandary:

I need to integrate this: (e^x + e^-x)/(e^x - e^-x) dx

Now, apparently the answer is ln(e^x - e^-x) + C. I'm not clear on why this is the answer. Anyone good at explaining the process between A and B? I have been told that (e^x + e^-x) is the derivative of (e^x - e^-x), but I thought it would simply be (e^x + e^-x). Why am I wrong?


Two more:

Integrate this definite integral (π is pi) : (b= π/4, a= π/2) cosΘ dΘ.

Supposedly that works out to sinΘ --> sin(π/4) - sin(-π/2)

Here, I'm not sure why that second π has become negative. I think it should be sin(π/4) - sin(π/2). Is there a rule I'm forgetting, or is this a typo?

Finally:

Integrate this definite integral (π is pi) : (b= π/2, a= 0) (sinΘ+cosΘ) dΘ

Supposedly that works out to (sinΘ) dΘ + (cosΘ) dΘ --> sin(π/2) - sin(0) + cos(π/2) - cos(0)

Here, shouldn't the cos(π/2) - cos(0) be negative? I was under the impression that when integrating sin... it becomes a -cos? Am I wrong here, or is this a typo?


Thanks! I'd appreciate any light being shed on this at all.

Starcross

For your first question (e^x + e^-x) is the derivative of (e^x - e^-x) because when you differentiate
e^(f(x)) for some function f of x you get f'(x)*e^(f(x)) where f'(x) is the derivative of f.

For example differentiating e^(x^2) gets you 2x*e^(x^2) and differentiating e^(-x) gets you -e^(-x) (as the derivative of -x is -1)

You've probably heard of this rule before. I think it's called the chain rule.

Loren Michael

Ah, awesome, thank you. That's completely clear.

One more question, actually, and this is one where I have no answer that confuses me, I just have no answer at all:

Integrate: e^x^3 dx

Is that possible? I know how to integrate (x^2)(e^x^3) dx and the rule involved (f'(x)e^(f(x))dx = e^(f(x))+C), but when I first saw the problem I didn't see the (x^2) and confused the hell out of myself for a little while.

EDIT: wait, isn't that just (3x^2)(e^x^3)? Jesus I'm dumb.

grungebox

The first one involves the chain rule, and you need to integrate by substitution. As for the other two, I'm pretty sure they're both typos.

Starcross

For the second problem I get sin(π/2) - sin(π/4). (As cos integrates to -sin)

For the third I get sin(π/2) - sin(0) + cos(0) - cos(π/2)

Starcross

Loren Michael wrote: »

Ah, awesome, thank you. That's completely clear.

One more question, actually, and this is one where I have no answer that confuses me, I just have no answer at all:

Integrate: e^x^3 dx

Is that possible? I know how to integrate (x^2)(e^x^3) dx and the rule involved (f'(x)e^(f(x))dx = e^(f(x))+C), but when I first saw the problem I didn't see the (x^2) and confused the hell out of myself for a little while.

EDIT: wait, isn't that just (3x^2)(e^x^3)? Jesus I'm dumb.

Your edit is the derivative of e^(x^3).

The integral of it probably doesn't exist using standard functions (ie. it can't be written down without using some obscure function you've never heard of)

grungebox

Loren Michael wrote: »

Ah, awesome, thank you. That's completely clear.

One more question, actually, and this is one where I have no answer that confuses me, I just have no answer at all:

Integrate: e^x^3 dx

Is that possible? I know how to integrate (x^2)(e^x^3) dx and the rule involved (f'(x)e^(f(x))dx = e^(f(x))+C), but when I first saw the problem I didn't see the (x^2) and confused the hell out of myself for a little while.

EDIT: wait, isn't that just (3x^2)(e^x^3)? Jesus I'm dumb.

Your edit is not correct. Are you sure that's the question? That's an extremely non-trivial integral that you can't really do by hand nor do I think it has a solution as an indefinite integral without some obscure math functions.

EDIT: According to MATLAB, the answer is:

-1/3*(-1)^(2/3)*(2/3*x*(-1)^(1/3)*pi*3^(1/2)/gamma(2/3)/(-x^3)^(1/3)-x*(-1)^(1/3)/(-x^3)^(1/3)*gamma(1/3,-x^3))

I'm not a math major, so I can't tell you much about gamma functions except that you shouldn't be seeing them at your level. I'm reasonably certain the question is not written the way it was intended (maybe it was asking for a derivative or something, or maybe it's just to try and fuck with you).

Loren Michael

Thanks all. You guys are great.