Substitution Integral Math problem

t = tan(x/2) is almost magic for solving trig function integrals. It is sometimes considered almost cheating. Anyway, ...

Why not start with t = tan(x/2) and the other two pieces of information you have been given?
 
In the integral to the right replace dx with 2/(1+ t^2) and sin x with 2t/(1+ t^2).

Now does this new integral make any sense?
 
Jomo said:
In the integral to the right replace dx with 2/(1+ t^2) and sin x with 2t/(1+ t^2).

Now does this new integral make any sense?
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Thank you for your response! I did do that, but the answer is still not correct. What am I doing wrong?
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tkhunny said:
t = tan(x/2) is almost magic for solving trig function integrals. It is sometimes considered almost cheating. Anyway, ...

Why not start with t = tan(x/2) and the other two pieces of information you have been given?
Click to expand...

Thank you for your response! I did start with the sin(x) and the dx parts, but I don't know how to use tan(x/2) as the professor never taught us in class. How would I go about using it in trig function integrals?
 
Ella Chen said:
View attachment 23360

I am really unsure of where and how to start. Any help is greatly appreciated!
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If you were me, you would probably want to start by checking the claims about dx and sin(x)! You'd find that they left out "dt" in the former. I never like to assume I've been told the truth ...

But if I were you, I'd realize that I don't need to do any of that. All I have to do is what Jomo said: make the substitutions and see what happens.

As for your supposedly incorrect answer, the next thing to do is, as they say, look over your work. Can you show it to us, so we can see if there is an error?

ADDED: Having now tried it, I see that your answer (possibly with a sign error) is really in terms of t, not x. You need to do a back-substitution.
 
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Ella Chen said:
Thank you for your response! I did do that, but the answer is still not correct. What am I doing wrong?
View attachment 23361
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You have not shared detailed work here. We cannot tell you 'what you are doing wrong" - unless you show us what you are doing!! Please post your detailed work.
 
Thank you so much everyone! I was doing it correctly, but I needed to switch the x to t and THEN substitute tan(x/2) for all the t's.
That was a weird one.
Appreciate the help!! :)
 
As pointed out by Dr Peterson and hinted by me there was a dt missing.
 
Ella Chen said:
Thank you so much everyone! I was doing it correctly, but I needed to switch the x to t and THEN substitute tan(x/2) for all the t's.
That was a weird one.
Appreciate the help!! :)
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As tkhunny mentioned, this is a substitution famous for its magical powers:

Tangent half-angle substitution - Wikipedia

en.wikipedia.org en.wikipedia.org

Quote from there: Michael Spivak wrote that this method was the "sneakiest substitution" in the world.

I imagine it was discovered by accident. I'd never think of it if I were trying to.

It's a little hard to remember the details of this powerful tool if you don't use it often, so I don't keep it in my back pocket, so to speak, but it's worth keeping on a shelf and knowing where to find it when you need it!

(I'll tell you a secret: When I tried the work, I myself accidentally wrote x instead of t when I factored after doing the substitution, so I easily recognized what you had done!)
 
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