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

Example of Use of Primitive of Reciprocal

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

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:

$\ds \ds \int_2^3 \dfrac {\d x} {x - 5}$

$=$

$\ds \ds \int_2^3 \dfrac {\d x} {5 - x}$

$\ds $

$=$

$\ds \bigintlimits {\map \ln {5 - x} } 2 3$

$\ds $

$=$

$\ds \ln 2 - \ln 3$

$\ds $

$=$

$\ds \ln \dfrac 2 3$

$\blacksquare$