Your question asked for the derivative of
y = arctan(3x/2)
so x = 2tan(y)/3
Use this substitution for the integral. It should simplify very nicely (assuming you did your differentiation correctly).
I've been doing some differentiation of inverse trig. In doing this I had to find the derivative of:
I found this to be:
To check this answer I thought I would try to integrate to see if I arrived back at the original question. However, I'm a bit stumped as to how I should go about integrating it?
I know that the original derivative was derived using the chain rule so is there some substitution that I can use to find the integral of
Thanks for the quick reply.
How would you find the right substitution if you were just given the question:
I'm sure that I could spot that it related to arctan (the x^2 and the positive constant is the give away) but how could I get it into a form from which I could integrate?
Sorry if I'm missing the obvious...
It is actually standard to try to substitute trig functions when you see forms like in the denominator of an integral (without an term in the numerator).
Normally you use the trig function that allows you to simplify the expression with a trig identity.
For example when you see in the denominator, you might use .
There may be multiple trig subs (or even hyperbolic subs, i.e. sinh, cosh) that work, but they should all come out to be equivalent answers.
I can see how you could reasonably easily work out which trig or hyperbolic to sub in. But given the integration question above, all I would have been able to tell you was that I should sub in tan. I wouldn't have been able to get the rest of the substitution at all...
Thanks for all the replies. They've been extremely helpful.
My only niggling problem is that in future I may be given slightly more complicated question in which the necessary substitution isn't quite so obvious. I think I'll always be able to identify which trig or hyperbolic I should use but is there a general way to work out what the correct full sub should be?
I've tried just subing in x = tan y. In the above example as a test to see if it's really always necessary to include in the correct coefficients and I haven't managed to make it work yet. (which may well be because my algebra is somewhat lacking...)
Look here for more info
x = some function of tan theta,
you might prefer to get the denominator in the form 1 + some square or other, thereby making it analogous to the left hand side of , which is the whole point of the sub. Then it's obvious to sub tan theta for directly for the square you made.
Just in case a picture helps...
Then adjust for this expression being a derivative via the chain rule
... where (key in spoiler) ...
Then you can make the direct analogy with integrating 1 + tan squared...
See also http://www.mathhelpforum.com/math-he...tml#post546765
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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