When to use trigonometric substitutions

[FlashStorm]

Hey,

Recently I studied trigonometric substitution way to solve many forms of Integrals. But since I'm new at this I can't get the intuition when to use what. When it goes out of normal "formulas" , I am really lost.

For example,
dx
S(----------- )
x^2*sqrt(x^2+1)I solved it, but had to use too many formulas for sinarctanx and cosarctgx (which are almost exact). anyway, At first I substituted x=tant, and later on I got:

S sin^-2(t)*cos(t)dt

and therefore i had to substitute z=sint

and in the end what I got is -1/sinarctgx+C.

(I checked it, if we will check what d(-1/sinarctanx) is equal to, you will get the original question, therefore its right)

And that's NOT A BEAUTIFUL FORM :(.

If can someone suggest something better out of his experience, Ill appreciate it.

Thanks in advance,
Aviv
But the way, I realized I can't copy from math-type to here (as expected, but that was my only guess).
Can someone tell me how write the function in more proper way?

Thanks

 

[itsjustme]

I came acroos something similar, here's how i got around the problem, i hope this helps:

 

[FlashStorm]

Rofl, I had this question one exercise ago, and it caused me great pain try doing it by integration by parts.
and now it looks so easy :P

Anyway I can't take any conclusions from your very nice solution to my problem. (since in my sqrt you can't just transform it to a sin/cos/tan), you can however transform it to cosh and sinh with the sinh2+1=cosh^2, But I am really weak with those functions so I am not touching them.

Anyway
Thanks , But it doesn't help me a lot :(

Any other suggestions?

 

[Office_Shredder]

Do you know what sin(arctan(x)) is? Try drawing a right triangle, label one side x, the other side 1, then try and find which angle is arctan(x). Finding sin(arctan(x)) becomes pretty simple after that