Question

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.

Answer 1

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

Answer 2

\[F(x)=\int\limits_{0}^{x^2+2x} (6+ \cos^2 t) dt\]
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
\[F'(x)= (2x+2)( 6+ \cos^2 (x^2+2x))-0\]

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

*problem