Primitive of Reciprocal/Examples/x minus 5 from 2 to 3

Example of Use of Primitive of Reciprocal

 * $\ds \int_2^3 \dfrac {\d x} {x - 5} = \ln \dfrac 2 3$

Proof
From Primitive of Reciprocal and using appropriate substitution:
 * $\ds \int \dfrac {\d x} {x - 5} = \ln \size {x - 5} + C$

$\dfrac 1 {x - 5}$ between $2$ and $3$ is negative.

So between $2$ and $3$:


 * $\ds \int \dfrac {\d x} {x - 5} = \map \ln {5 - x} + C$

Hence: