Question

What is the integral of `int (2 -1/x)dx` from 1/2 to 3?

Answer 1

`5-ln(6)`

Explanation:

Integrate each of these by part. Note that `d/dx(2x)=2`, so `int2dx=2x`, and `d/dx(ln(x))=1/x`, so `int1/xdx=ln(x)`:

`int_(1/2)^3(2-1/x)dx=[2x-ln(x)]_(1/2)^3`

`=[2(3)-ln(3)]-[2(1/2)-ln(1/2)]`

`=6-ln(3)-1+ln(1/2)`

Note that `-ln(3)+ln(1/2)=-ln(3)-ln(2)=-(ln(3)+ln(2))=-ln(6)`:

`=5-ln(6)`