Please can someone help me to integrate:
sin^3x
INT(sin^3 x)dx= (-1/3)(cos x)(sin^2 x + 2)
Oh wait, did you want me to show my work???
Write sin^3x as (sin x)(sin^2x) and use sin^2x + cos^2x = 1 to write this as (sin x)(1 - cos^2x) which equals
sin x - sin x cos^2x
Now sin x integrates to -cos x and (by the chain rule) sin x cos^2x integrates to -1/3 cos^3x. So the whole thing integrates to
-cos x + 1/3 cos^3x +c
I always forget the obligatory +C
That’s a mark off where I come from 
Yea, here too!! 
Sorry to be stupid but i don’t get this bit.
Differentiate cos^3x using the chain rule. In full
y = cos^3x = (cos x)^3 = u^3 where u = cos x
Then dy/du = 3u^2 and du/dx = -sin x
By the chain rule, dy/dx = dy/du * du/dx = -3u^2 sin x = -3sin x (cos x)^2
i.e. d/dx {cos^3 x} = -3sin x cos^2 x
Integrate both sides: INT {-3sin x cos^2x} = cos^3x
Divide by -3: INT {sinx cos ^2x} = -1/3 cos^3x
This splitting up the integrand is a standard trick and will work for any odd power of sin x or cos x.
According to the Standard Mathematical Tables from the Chemical Rubber Publishing Company the integral of sin[sup]3[/sup] x = -1/3cos x(sin[sup]2[/sup] x + 2) + const.
Y’know, I would feel a lot better about this thread if I could think of any possible reason, other than a homework assignment, why the OP would want to know this.
Yeah, chronos you were right, it is for homework. Unfortunatley the integrating skills that we have been taught up to date have not allowed us to handle this sort of integrating. I have no idea why I got asked this question in the homework!
David, where do you think I got my answer???
Ah, but his had the +const.
I guess that means Clarissa’s using the Chemical Rubber Publishing Company’s Nonstandard Mathematical Tables.
If you’ve a keen eye, you can differentiate between them.
I didn’t notice that you had already posted the correct answer. I had no intention of working the problem the hard way, so I started the post then went looking for my book. Clever eh? By the time I found it you beat me to it.
I opened this thread all set to discuss multiracial neighborhoods.
Zut has it! ‘Nonstandard’ indeed!
>> If you’ve a keen eye, you can differentiate …
We’re trying to integrate which is pretty much the opposite.