Question

I need some help with Trigometric substitutions, can someone help me?

Answer 1

Depends... post your question, we'll see :)

Answer 2

\[\int\limits \frac{ 1 }{ \sqrt{1-x ^{2}} }dx\]

Answer 3

Okay, whenever you see a form
\[\Large \sqrt{1-x^2}\] particularly in the denominator, think about using the substitution
\[\Large x = \sin(\theta)\]

Answer 4

ok

Answer 5

So, it follows that...
\[\Large dx = \cos(\theta)d\theta\] yes?

Answer 6

yes

Answer 7

Use that substitution for.
\[\Large \sqrt{1-x^2}\]
replace x with \(\sin(\theta)\) and tell me what you get...

Answer 8

\[\sqrt{1-\sin(\theta)^{2}}\]

Answer 9

Let's have it as
\[\Large \sqrt{1-\sin^2(\theta)}\]
I'm sure that's what you meant, now what is this equal to?

Answer 10

Remember that
\[\Large \cos^2(\theta) + \sin^2(\theta) = 1\]

Answer 11

ok

Answer 12

So...? What is
\[\Large \sqrt{1-\sin^2(\theta)}=\color{red}?\]

Answer 13

\[\cos ^{2}\theta\]

Answer 14

tsk tsk :P
\[\Large = \sqrt{1-\sin^2(\theta) }={\LARGE \color{red}{\sqrt{\color{black}{\cos^2(\theta)}}}}\]

Answer 15

Don't forget the radical :P

Answer 16

So anyway...
\[\Large \frac1{\sqrt{1-x^2}}=\frac1{\sqrt{1-\sin^2(\theta)}}=\frac1{\cos(\theta)}\]
ok?

Answer 17

this would be the final step right?

Answer 18

Almost, we also have to deal with the dx...
\[\Large \int \frac1{\cos(\theta)}dx\]but we have an expression for dx in terms of \(d\theta\)

Answer 19

+c

Answer 20

No...
remember this?
\[\Large x = \sin(\theta)\]
so that...
\[\Large \frac{dx}{d\theta}=\cos(\theta)\]

Answer 21

\[\int\limits \frac{ 1 }{ \cos \theta}\cos \theta\]

Answer 22

smh :3
\[\Large dx = \cos(\theta)d\theta\]
\[\Large \int \frac1{\cos(\theta)}dx = \int \frac1{\cos(\theta)}\cdot \cos(\theta)d\theta\]

Answer 23

I trust you can do it from here?

Answer 24

still not sure about the next steps

Answer 25

Well, the cos cancels out like this...
\[\Large \int\frac1{\cancel{\cos(\theta)}}\cancel{\cos(\theta}d\theta=\int d\theta\]
and this is possibly the simplest integral there is :P

Answer 26

okay i forgot about crossing out the cos

Answer 27

for the 1 up top would i put that out side of the interval? or because it's a constant do I get rid of it?

Answer 28

Uhh, sure, but 1 times any number is just that number, so the 1 just disappears? -.-