# Tag Archives: 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 \\&= […] # Integration of Rational Functions 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 […] # Must-Know 10 Basic Translations of Rational Functions Explained Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote. Any graph of a rational function can be obtained from the reciprocal function f(x)=\dfrac{1}{x} by a combination of transformations, including a translation, stretches and compressions. Type 1: Vertical Compression \( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 The […] # Indefinite Integral of Rational Functions

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 […]