Question

dy/dx = tany / x
I need to find the anti-derivative of that, pwease help :3

Answer 1

use separation of variables.

Answer 2

separate the variables and integrate

Answer 3

mmm how would I do that? :/
like dy/dx = (tany)(1/x) ?

Answer 4

same process used in the previous problem

Answer 5

really? but how would I do dy/tany = dx/x ? :/

Answer 6

\(\sf \frac{dy}{dx}=\frac{tan(y)}{x} \Rightarrow \frac{1}{x}dx=\frac{1}{tan(y)}dx\)

Answer 7

sorry, the [(tan(y)]\(^{-1}\) should be dy, not dx

Answer 8

1/tany = coty
dx/x = (1/x)dx,

so what is the integral of cot(y) and what is the integral of 1/x?

Answer 9

yes.

Answer 10

mmmm -1 / (1+ y^2) = lnx ? :3

Answer 11

what?

Answer 12

the anti derivative?
coty dy = -1 / (1+y^2) right?

Answer 13

no wait, nvm that would be arccot >,<

Answer 14

No.

Answer 15

lol xD
ignore what I was saying before :P

ln|siny| = lnx + C ?

Answer 16

right

Answer 17

soooo siny = e^(lnx + C) -> siny = C x ?

Answer 18

right

Answer 19

then y = arcsin(Cx) ??? :/

Answer 20

right

Answer 21

That's all? Thank you ^_^

Answer 22

\[\int\limits \cot y dy=\int\limits \frac{ \cos ~y }{ \sin ~y }dy=\ln \left| \sin y \right|\]