Question

Find the indefinite integral of: [1 + 4cotx]/[4 - cotx] dx

I would like a hint please

I asked the last time and someone said to use the u substitution of of tan(x/2). However, I do not know how they got tan in the first place. All I see is cot

Answer 1

I feel like I wanna start with\[\int{1+4\cot x\over4-\cot x}dx=\int{\sin x+4\cos x\over4\sin x-\cos x}dx\]but it's just a hunch...

Answer 2

That is exactly what I did too

Answer 3

oh wait, that should do it :)

Answer 4

And then I did not know where to proceed from there

Answer 5

\[u=4\sin x-\cos x\]\[du=4\cos x+\sin xdx\]

Answer 6

WOW

Answer 7

I had no idea

Answer 8

Is there not another way to do this?

Answer 9

just gotta try it sometimes ;)
um... I don't know this is the first one that comes to mind, and it's pretty easy

Answer 10

Ok thank you for your time

Answer 11

I bet there is a wolfram-type way to do it that starts with u=tan(x/2) (which is where I suspect the earlier advice you got came from) but that would get very ugly I bet...

Answer 12

Yea

Answer 13

yeah if you want to make your life impossible do it this way
http://www.wolframalpha.com/input/?i=integral(1%2B4cotx)%2F(4-cotx)dx&t=crmtb01
(click show steps, and notice how the person who previously advised you on that sub u=tan(x/2) probably couldn't do the integral themselves, and tried to look smart by quoting the wolf, lol)

Answer 14

L.O.L