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