Trigonometric substitution?

Question

I was reviewing the integration technique of trigonometric substitution and I wanted to know why we are only restricted to using sin(a), tan(a), and sec(a)....

For example, in order to find the integral of (1/(1-x^2))dx .... the common method is to use x = sin(a) as the subsitution and you then get the answer arcsinx + C. However, how come you can't use x = cos(a)??? I can work out the problem but I get a different answer. I don't have my notes with me at the moment, but I believe I got -arccosx + C.

Why must I use sin(a)???

Answer 1

Note that:
arc(sin x) + arc(cos x) = (pi)/2
So, arc(cos x) = (pi)/2 - arc(sin x).

Your answer with cosine substitution is:
-arc(cos x) + C
= -[(pi)/2 - arc(sin x)] + C
= arc(sin x) + (C - pi/2)
= arc(sin x) + another constant, say C1.
= answer with sine substitution

Thus both answers are same.

Well, well, well; either you have typed the integral incorrectly or you have done the integration wrongly because the correct answer to the integral you have give is not that you provided. The integral [1/sqrt(1 - x^2)] dx gives the answer you provided. So you might have missed the square-root sign.

By the way, the result you have attained is valid only for |x| < 1. So trigonometric substitution can be used.

Answer 2

Actually, your question is valid and the answer is that choosing between sine and cosine for example, is pretty much arbitrary. The reason sine substitution is preferred is because if you substitute cosine, you get a minus sign, so using the recommended substitutions (sine, tangent, secant) you get an integral which is slightly simpler.

Why don't you just try it on your own?
Do a problem with sine substitution and then do the same exact problem with a cosine substitution and see the difference yourself.

Answer 3

It's not mandatory, it's just been found that if you use the recommended substitutions, you have a better chance of getting a nice clean answer without too much pain.

Answer 4

Edited: (before I had made the blunders and positioned integration of one million/?(a million-4x²). i realized this mistake on revisiting the question) ??(a million-4x²) dx enable 2x = sin? => 2dx = cos? d? ??(a million-4x²) dx = (a million/2) ? cos? * ?(a million - sin²?) d? = (a million/4) ? 2cos²? d? = (a million/4) ?(a million + cos2?) d? = (a million/4) (? + (a million/2)sin2?) + c = (a million/4) arcsin(2x) + (a million/8) * 2sin? cos? + c = (a million/4) arcsin(2x) + (a million/2) x?(a million-4x²) + c