Question

Calculate the following derivatives
\[\Large \frac{d}{dx} \int_0^x dt \cos(t^{3}+1)\]

Answer 1

wow what class is this for?

Answer 2

this is for college second semester

Answer 3

.... if you interate and then take the derivative you go back to the original function

Answer 4

oh then the answer is probalby

Answer 5

yeah what he said!

Answer 6

but change the t to an x

Answer 7

exactly

Answer 8

i have already answer for it, but its too short, i want to see in othwer ways..

Answer 9

well you sure as hell don't want to integrate that. I tried for about a minute and It seems crazy. then I went back and saw the d/dx. No Idea how you would show your work anymore than that

Answer 10

here is the solution but its too short to understand it..
\[\frac{d}{dx} \int_0^x f(t) dt=f(x)\text{ by the FTC, so }\\
\frac{d}{dx} \int_0^x \cos(t^{3}+1) dt=\cos(x^{3}+1)\]

Answer 11

Alright so tunahan, Derivatives and Anti Derivatives are transformations. if you take the anti derivative of f(t), you'll get F(t) correct?

Answer 12

yes..

Answer 13

what this is saying is thta if you take the antiderivative you'll get F(t)... only your integration limits are x and 0 so you replace the t with x because

F(x)-F(0)=F(x)

Answer 14

so now you have function of x correct

Answer 15

well you are integrating it with respect to t, but your limits of integration will give you a function of x minus some constant right? so now you take the derivative of that and the costant part goes away so you are just taking the derivative of the antiderivative which just is what you had before

Answer 16

when you take the derivative you get f(x) . which is the same as f(t) only the variablesare switched

Answer 17

is it possible to have a larger solution for same question ?

Answer 18

if it's possible to find a solution to the integral

Answer 19

ok..

Answer 20

ok thank you both guys..

Answer 21

but it's not necessary... if the integral is easy you can actually evaluate it and you'll get the samt hing

Answer 22

This is just a fundamental theorem

Answer 23

http://www.wolframalpha.com/input/?i=cos%28t%5E3%2B1%29dt

Answer 24

this is why you want to understand the short way because it'd be nearly impossible to solve that

Answer 25

ok i think i need to work more on integrals and derivatives...and then ask questions it will help me more to understand.. i have tomorrow exam, i dont know anything =)

Answer 26

wow its really hard question according to wolfram =)

Answer 27

that's why I asked you what level math you were in at the beginning. I only saw that integral and thought you needed to evaluate it. It's a really easy question but if you just had to find that without the d/dx in front it would be a really hard one...