What does it mean when I'm asked to solve an integral of a function of t between within a section with borders in term of x?
See attached question for example.

Question

anonymous · Answer

attachment

ash2326 · Answer

What you have to do is find F(x) first then differentitate it ,
when you integrate the function in t  and  we are given the limits in x , so it'll  become a function in x
You can use the method to easily evaluate F'(x) without integrating

ash2326 · Answer

$$F(x)=\int\limits_{0}^{x^2+2x} (6+ \cos^2 t) dt$$
let's differentitate with respect to x
$$F'(x)= [6+\cos^2 (x^2+2x)](d/dx (x^2+2x))-[6+\cos^2 (0)](d/dx (0))$$
substitute upper limit in t and product of this with the derivative of upper limit - the lower limit in function and product of it with the lower limit's derivative .
you'll get
$$F'(x)= (2x+2)( 6+ \cos^2 (x^2+2x))-0$$

ash2326 · Answer

sorry it was cos t^2, then you'll get
$$F'(x)= (2x+2)(6+ \cos (x^2+2x)^2)$$

anonymous · Answer

Ok I think I understand. Thanks a lot. I've been trying to do this all day.

ash2326 · Answer

no problem, just remember the method, you'd be able to solve any proble like this

ash2326 · Answer

*problem