1. ## Integration question

Hi

let g(x) =

where f is an increasing function. Then,

a. g{x} is an increasing function of x.
b. g{x} is an decreasing function of x.
c. g{x} is an increasing function of x>0 and decreasing for -10<x<10.
d. none of the above.

thanks

2. g ' (x) = x f '(x) (from 2d fundamental theorem)

since f(x) is increasing f ' (x) > 0

Therefore the answer is increasing if x > 0 and decreasing if x < 0

Are you sure c is Right it doesn't make sense seems it should read:

increasing x>0 and decrasing -10< x < 0

3. hi!
it depands on what kind of integration that you consider,if you consider the lebesgue integration, then you can choose f as a Cantor function,then f is increasing function,while f'=0almost everywhere.so g=0forever!

4. thanks Can the Second Fundamental theorem be applied to Definate integrals also? I was under the impression that it was only for indefinite ones.

5. and if you conside Riemann integration,just like what Calculus26 said!

6. Originally Posted by berlioz
hi!
it depands on what kind of integration that you consider,if you consider the lebesgue integration, then you can choose f as a Cantor function,then f is increasing function,while f'=0almost everywhere.so g=0forever!
i dont think we have done other types of integrals.

7. ## 2d fundamental thm

See attachment for statement of 2d FTC