Can You Use Simple Substitution for Tricky Integrals?

[climbhi]

If you had say ∫cos⁴(x)dx according to my integration table in calc book this would be something nasty. Could you not say let u = sin⁵(x)/5 therefore du = cos⁴(x)dx and then ∫du = u = sin⁵(x)/5 + C. Is there something wrong with this. This technique would work on ∫x² if you said let u = x³/3 and then did everything else the same except there its not quite so tricky. I guess what I'm asking is if you're good at designing a function that when differentiated would give the funtion in the integral can you use my method there instead of the tables which give this nasty formula: ∫cosⁿ(x)dx = [(cos^n-1x)(sinx)]/n + [(n-1)/n]∫cos^n-2(x)dx

 

[pi-70679]

The problem with what you are trying to say is that your basis is totally false. The derivative of 1/5 sin^5(x) is not cos^4(x), but sin^4(x)cos(x). This is by the chain rule. You should remember that the reason why substitution eists is precisely because not everything can be treated as simply x and certainly not trig functions! I think it would be good for you to review how to do derivatives, and if you're stuck and want to check the answer for an integral, try and derive it first to get back to the original equation you just integrated. This way you're sure it's right, and derivatives are safer to do than integrals usually.

 

[climbhi]

[beats self relentelessly on head] Oh man I feel soo stupid, I cannot believe I missed that! I knew it was way to easy that way. I can't even describe how stupid I feel looking over that. Ohh well what can you do?[/end self beating blushing terribly]

 

[pi-70679]

Don't worry about that, i did much worse in an exam situation:
I was extremely stressed because the exam was simply too long, as i rushed in the last question, i accidentaly derived instead of integrating, in an eletric field question. Just image how ashamed i was when i got my paper back. The teacher thought i was a total idiot, even though he was forced to change his mind later on when my average got back above 90%.