Consider an integrable function f(x) that is odd

Question

Assume that integral from 0 to 5 f(x)dx=6 and integral from 0 to 8 f(x)dx=4

 

Find:

-integral from 5 to 8 f(x)dx

-integral from -5 to 8 f(x)dx

 

                   

 

Please Help!!

Answer 1

∫f(x)dx from 5 to 8 

= [ ∫f(x)dx from 0 to 8 ] - [ ∫f(x)dx from 0 to 5 ] 

= 4  -  6  

=  -2 

 

f(x) odd  ==>  f(-x) = -f(x)  ==> 

[∫f(x)dx from -5 to 0]  =  - [∫f(x)dx from 0 to 5]  =  -6  

 

∫f(x)dx from -5 to 8 

= [ ∫f(x)dx from -5 to 0 ] + [ ∫f(x)dx from 0 to 8 ]

=  -6  +  4

=  -2