Integrating trig functions

[Davidm]

Is there any reason why the integral of simple trig function like:

Sin X raised to the power N is not (1/N+1)SinX raised to N+1*derivative of SinX (being CosX)?

I do know and follow the methods used to resolve the problem, including the use of integration by parts. But I am confounded to find why the basic integration approach using the chain rule does not work in this case.

 

[tkhunny]

Just some thoughts:
1) You can always resort to the definition of the Riemann Integral. You will find satisfaction.
2) Because it doesn't?
3) Why would it be the derivative? Don't you mean "antiderivative"? (Which is funny, because it sort of the same thing in this case.)
4) Integration by Parts is a variation on the Product Rule. Why do you want the Chain Rule in there?

 

[Dr.Peterson]

The chain rule applies to differentiation, not integration. The latter is the inverse of the former. Nothing in integration is quite as simple as in differentiation.

But the chain rule, turned inside out in integration, becomes the method of substitution.