Question

Integrate : ∫dx/(sqrt((x^2)-16))

Answer 1

If you remember your pythagorean identities, you can ALWAYS tell.

sin^2(x) + cos^2(x) = 1

Divide by cos^2(x)

tan^2(x) + 1 = sec^2(x)
or
tan^2(x) = sec^2(x) - 1

Divide by sin^2(x)

1 + cot^2(x) = csc^2(x)
or
cot^2(x) = csc^2(x) - 1

The one it looks most like is the one you pick. In this case, x^2 + 4^2 means we'll need this one tan^2(x) + 1 = sec^2(x). Multiply by 4^2.

Answer 2

did you just copy and paste from this previous question lol http://openstudy.com/study#/updates/5085e59de4b0b7c30c8e3fa5

Answer 3

i already checked it out still dont get it :(

Answer 4

Yes I did lol But I know it works as this is a simple technique I mostly use in pythagorean identities

Answer 5

First off, You must have your integral be something like this: 1/sqrt(1+u^2) where it can be anything so it is easier this way.

Answer 6

yea but how do you get to that point, that all i wanna know

Answer 7

Do you need the formula step-by-step or just the answer itself with an explanation?

Answer 8

just how to get to something like 1/sqrt(1+u^2)

Answer 9

when you have the form \(\Large \sqrt{x^2-a^2}\) in the denominator, put
\(\Large x= a\sec t \)
so, here, put x =4 sec t
dx = ... ?

Answer 10

4sec t tan t ?

Answer 11

yes,
and your denominator now becomes ?

Answer 12

i dont get it bro, like should i substitute this in denominator ?

Answer 13

yes,
\(\sqrt{x^2-a^2 }= \sqrt {a^2\sec^2t-a^2} = ... ?\)
ccan you simplify ?

Answer 14

a sec t - a

Answer 15

nopes...

Answer 16

\(\sqrt{x^2-a^2 }= \sqrt {a^2\sec^2t-a^2} = \sqrt{a^2(sec^2t-1)}=a \sqrt {\tan^2 t}=a\tan t\)
got this

Answer 17

oh and here a=4

Answer 18

i got it buy that i should put x=a sect ? , is it a formula ?

Answer 19

we put x = a sec t just so that denominator would simplify to a tan t!
like if we had ,
sqrt (a^2-x^2) we would have put x = a sin t , so that denominator would become a cos t!

Answer 20

if it was (a^2-x^2) i would simple use arcsin(a/x) + c formula but anyways, now the denominator becomes 4 tan t , how does that helps

Answer 21

notice that numerator is 4 sec t tant
what gets cancelled ?

Answer 22

we are left with sec t

Answer 23

yes, just integral sec t dt
can you integrate that ?

Answer 24

ln |sec t + tan t| + C ?

Answer 25

correct!
now just plug back in
t = sec inverse x

Answer 26

sorry,
t = sec inverse (x/4)

Answer 27

because x = 4 sec t

Answer 28

no...
ln |x/4 + tan (sec inverse (x/4))| +c

do you know how to simplify
tan (sec inverse (x/4)
?

Answer 29

plz explain how u got 4 sec t tant in the numerator and how i got cancelled