Question

alright, i know that the integral of 1/(1+x^2)dx is arctanx, but can someone please help me understand why?

Answer 1

youd have to work it backwards from the derivative

Answer 2

\[tan^{-1}(x)=y\]
\[x=tan(y)\]
\[1=sec^2(y)\ y'\]
\[\frac{1}{sec^2{y}}= y'\]

Answer 3

since tan(y) = x

tan^2(y)+1 = sec^2(x)

1/x^2+1

Answer 4

barring typos of course ;)

Answer 5

Integral is also known as ante-derivative hence the above is true..

also integral of 1/(1+x^2)dx is *not* arctanx, you missed the constant which means a family of functions.

Answer 6

right, yes my bad arctan+C. but if i saw 1/(1+x^2) on my test and had to integrate it and did not remember that it was arctan+C, there must be a way to figure it out right?

Answer 7

well, since x^2 + 1 is not decomp across the reals .... maybe something to do with a complex trick

Answer 8

or, maybe a usub into x = tan(u) type thing

Answer 9

but then what would you do with a 1/sec^2(u) ...

Answer 10

ok, i do think that i am supposed to substitute tanu=x, only because someone else told me to.

Answer 11

cos^2(u) has no sin(u) to work with ....

Answer 12

but then you have 1/1+tan^2u (sec^2u)

Answer 13

the hard part is trying to get to tan-1 without going into complex definitions; and I dont think that is possible at the moment

Answer 14

and those cancel out and just give you integral du....but i dont think that is right?

Answer 15

its not right ...

Answer 16

@amistre64:It's not that hard just think right angle triange :)

Answer 17

triangles are pointy, and hurt lol

Answer 18

haha, alright. so maybe i will just have to accept that i need to memorize it :)

Answer 19

but i think the caveat is to manipulate it without remembering stuff

Answer 20

its not hard to memorize :) its one of the basics for these things

Answer 21

No need to memorize .. you may think of tangents of hyperbola if you want .. geometrical understanding is very essential in calculus :)

Answer 22

memorize it fer sure; you might need to forget a little algebra to fit it in, but its worth it in the long run ;)

Answer 23

states and their capitals are pretty useless; might try fitting it in there

Answer 24

@amistre64:Memorization is bad ...... it's makes you memory dependent :P

Answer 25

states capital ??????

Answer 26

LOL i am to the point where i have to forget other things to fit new math in my brain. But ok, i wish i had the geometric understanding to see it...

Answer 27

tangents of hyperbola? im not really sure what you mean

Answer 28

sinh, cosh, and .... tanth?

Answer 29

\[sinh = \frac{e^{-x}+e^x}{2}\]
maybe

Answer 30

x^2-y^2 = a^2 is a hyperbola ..

Answer 31

Hm, alright. i know what a hyperbola is but im not sure what to do with that. It's ok tho i will continue to study. Thank you! :)