Use the fundamental theorem of calculus to findd/dx integral from (-x)^x [z-1]/[z+2] dz (-x) is suppose to be on the bottom and (x) is suppose to be on the top.   Also, please split the integral to two integrals each of which will have only one limit that varies. I geuss you can call it the early stages of the Fundamental Theorem. We haven't learned more about it yet.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You need to evaluate the definite integral using the linearity of integral such that:

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x z/(z + 2)dz - int_(-x)^x 1/(z+2) dz`

`int_(-x)^x z/(z + 2)dz = int_(-x)^x (z+2-2)/(z + 2)dz`

`int_(-x)^x z/(z + 2)dz = int_(-x)^x (z + 2)/(z + 2)dz - int_(-x)^x 2/(z + 2)dz`

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x (z + 2)/(z + 2)dz - int_(-x)^x 2/(z + 2)dz - int_(-x)^x 1/(z+2) dz`

`int_(-x)^x (z - 1)/(z + 2) dz = int_(-x)^x dz - int_(-x)^x 3/(z + 2)` `dz`

` int_(-x)^x (z - 1)/(z + 2) dz = z|_(-x)^x - 3ln|z + 2||_(-x)^x`

`int_(-x)^x (z - 1)/(z + 2) dz = (x + x) - 3(ln|x+2| - ln|2-x|)`

`int_(-x)^x (z - 1)/(z + 2) dz = 2x - ln|(x+2)/(2-x)|^3`

Hence, evaluating the definite integral yields `int_(-x)^x (z - 1)/(z + 2) dz = 2x - ln|(x+2)/(2-x)|^3.`

Approved by eNotes Editorial Team