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

 Integration of $\displaystyle \dfrac{1}{x}$ $$ \large \begin{align} \displaystyle\dfrac{d}{dx}\log_ex &= \dfrac{1}{x} \\\log_ex &= \int{\dfrac{1}{x}}dx \\\therefore \int{\dfrac{1}{x}}dx &= \log_ex +c\end{align} $$ Example 1 Find $\displaystyle \int{\dfrac{2}{x}}dx$. \( \begin{align} \displaystyle\int{\dfrac{2}{x}}dx &= 2\int{\dfrac{1}{x}}dx \\&= 2\log_ex +c\end{align} \) Example 2 Find $\displaystyle \int{\dfrac{1}{3x}}dx$. \( \begin{align} \displaystyle\int{\dfrac{1}{3x}}dx &= \dfrac{1}{3}\int{\dfrac{1}{x}}dx \\&= \dfrac{1}{3}\log_ex +c\end{align} \) Example 3 Find $\displaystyle \int{\dfrac{4}{5x}}dx$. \( \begin{align} \displaystyle\int{\dfrac{4}{5x}}dx […]

Understanding the Indefinite Integral of Rational Functions Using Indefinite Integral of Rational Functions requires that the format of the expression must be the power of linear expressions, such as \( (3x-1)^3, (2x+3)^3, \sqrt{4x-1} \), etc. \( (3x^2-1)^3, (2\sqrt{x}+3)^3, \sqrt{4x^3-1} \) are not applicable for this formula.$$ \large \displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \ (n […]