Definite Integral of Rational Functions

$$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e {f(x)}\big]_{n}^{m} \\ &= \log_e{f(m)} - \log_e{f(n)} \end{align} $$

Example 1

Find $\displaystyle \int_{1}^{2}{\dfrac{1}{x}}dx$.

Example 2

Find $\displaystyle \int_{1}^{3}{\dfrac{2}{2x-1}}dx$.

Example 3

Find $\displaystyle \int_{0}^{2}{\dfrac{1}{3x+2}}dx$.

Example 4

Find $\displaystyle \int_{1}^{3}{\dfrac{2}{5x-1}}dx$.

Example 5

Find $\displaystyle \int_{3}^{5}{\dfrac{x}{x-2}}dx$.

Example 6

Find $\displaystyle \int_{0}^{2}{\dfrac{2x-1}{x+3}}dx$.