Question

How do you evaluate `int 1/(x-3)dx` from 1 to 4?

Answer 1

`-ln2`

Explanation:

If we let `u=x-3`, then `du=dx`.

This transforms the integral as follows:

`int_1^4 1/(x-3)dx=int_-2^1 1/udu`

The transformation of the bounds on the definite integral occurs since we switched variables (from `x` to `u`). To find the values they switch to, plug in `1` and `4` into `u=x-3`.

We see that `int1/udu=lnabsu+C`, so we can evaluate the integral:

`=[lnabsu]_-2^1=lnabs1-lnabs(-2)=0-ln2=-ln2`