Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

Inverse trig functions

Hello, I'm back again after a while, with a question that has me stumped.

I'm doing an exercise on integration by substitution and was asked to intergrate:

Using the substitution:

So, after doing the integral, I ended up with:

I think, judging by wolframalpha, this is probably not absolutely correct. I allows myself to cheat at one point, for example, by saying:

Which obviously is not true for all values of θ, but this is an A-level textbook I'm using which here is trying to teach integration by substitution, so I assume it expects me to do this.

So obviously I then need to give the answer in terms of x, which is where things go a little wrong. This is what I've done:

Which obviously gives me a final answer for the integral of:

But the book gives:

This seems so close to what I have that I think I must be missing something, but I can't work out what. Wolframalpha says that the two aren't equivalent, so I don't quite know why the two answers are different.

Last edited by Au101 (2014-08-13 05:39:13)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Inverse trig functions

hi Au101,

Welcome back.

Your working all looks good to me.  So I tried the two functions in the MIF function grapher:

http://www.mathsisfun.com/data/function … =acos(2-x)

They only appear to differ by a constant; which makes them 'the same' for an indefinite integral.  So I think you can stop worrying. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: Inverse trig functions

Indeed:

[list=*]
[*]

taking the range of arccos to be

[/*]
[/list]

Proof:

[list=*]
[*]

[/*]
[/list]

Thus

in the interval

.

Last edited by Olinguito (2014-08-14 11:15:07)

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Bassaricyon neblina

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Inverse trig functions

Ahhhh thank you both!