Tag Archives: Rational Functions

Definite Integral of Rational Functions

Definite Integral of Rational Functions

$$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{n}^{m} \\ &= \log_{e}{m} – \log_{e}{n} \end{align} $$ Generally, $$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e{f(x)}\big]_{n}^{m} \\ &= \log_{e}{f(m)} – \log_{e}{f(n)} \end{align} $$ 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 […]

Integration of Rational Functions

Integration of Rational Functions

Integration of $\displaystyle \dfrac{1}{x}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\log_ex &= \dfrac{1}{x} \\ \log_ex &= \int{\dfrac{1}{x}}dx \\ \therefore \int{\dfrac{1}{x}}dx &= \log_ex +c\\ \end{align} $$ 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} […]

Must-Know 10 Basic Translations of Rational Functions Explained

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: Horizontal Compression \( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 \) The […]

Indefinite Integral of Rational Functions

Indefinite Integral of Rational Functions

Understanding Indefinite Integral of Rational Functions Using Indefinite Integral of Rational Functions requires that the format of the expression must be 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. $$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \ (n \ne -1)$$ Practice […]