# Definite Integral of Rational Functions

\large \begin{align} \displaystyle \int_{n}^{m}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{n}^{m} \\ &= \log_{e}{m}-\log_{e}{n} \end{align}
Generally,
\large \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 $\displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx$.

\begin{align} \displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx &= 2\int_{1}^{5}{\dfrac{1}{x}}dx \\ &= 2\big[\log_{e}{x}\big]_{1}^{5} \\ &= 2 \log_{e}{5}-2 \log_{e}{1} \\ &= 2 \log_{e}{5}-2 \times 0 \\ &= 2 \log_{e}{5} \end{align}

## Example 2

Find $\displaystyle \int_{2}^{8}{\dfrac{3x}{x^2+1}}dx$.

\begin{align} \displaystyle \int_{2}^{8}{\dfrac{3x}{x^2+1}}dx &= \dfrac{3}{2} \int_{2}^{8}{\dfrac{2x}{x^2+1}}dx \\ &= \dfrac{3}{2} \int_{2}^{8}{\dfrac{(x^2+1)’}{x^2+1}}dx \\ &= \dfrac{3}{2} \big[\log_e{(x^2+1)}\big]_2^8 \\ &= \dfrac{3}{2} \big[\log_e{65}-\log_e5\big] \\ &= \dfrac{3}{2} \log_e{\dfrac{65}{5}} \\ &= \dfrac{3}{2} \log_e{13} \end{align}