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} […]

Transformation of Rational Functions

Transformation of Rational Functions

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. $y=\dfrac{a}{x}$: horizontal compression, $0 \lt a \lt 1$ $y=\dfrac{a}{x}$: horizontal stretch, $a \gt 1$ […]

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 […]