Question

Question inside: Calculus

Answer 1

\[p(x) = \int\limits_{-1}^{x^3} (t^4+1)^7\]

Answer 2

what is derivative of p(x)?

Answer 3

So this is one of the fundamental theorems of calculus (the 1st or 2nd one, I forget). But when you have a problem like this, it's actually quite simply. The first thing you want to do is replace all the t's in your problem with the variable that appears in the integral limits. So basically replace t with x^3. After that, you multiply by the derivative of that variable expression. So in this case you want to multiply by the derivative of x^3. After that you're done. The only note to make is if your variable expression is on the bottom limit of the integral. If this is the case, your answer needs to be made negative. So to summarize:

1. Replace all t's with the variable expression in the limit:
\[((x^{3})^{4}+1)^{7}\]
2. Multiply by the derivative of the variable expression:
The derivative of x^3 is 3x^2, so I get:
\[3x^{2}(x^{12} +1)^{7}\]
3. If the variable expression was the bottom limit, multiply by -1. Of course it was on top, so we dont need to do that. The above is you derivative :)

Answer 4

Well, I understand the process, but I don't understand why you do that. I will learn that at a later time though. Thanks for your help.

Answer 5

Well, what essentially happens is you take the integral like normal, get some answer, then take the derivative of said integral answer. It's just that process is longer if not impossible by hand depending on the integral. So in this specific circumstance you just would use this theorem that let's you do this. If you actually expanded that (t^4+1)^7 out and did all the integration, term by term, plugged in your limits, and then finally took the derivative of your answer, that entire mess would simplify to what I have above. Just yeah, there's no reason to go through the trouble :P

Answer 6

Oh right. It's the fundamental theorem of calculus. Okay thanks again

Answer 7

Np :)