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 &= \dfrac{4}{5}\int{\dfrac{1}{x}}dx \\ &= \dfrac{4}{5}\log_ex +c \end{align}

Example 4

Find $\displaystyle \int{\dfrac{6}{-7x}}dx$.

\begin{align} \displaystyle \int{\dfrac{6}{-7x}}dx &= -\dfrac{6}{7}\int{\dfrac{1}{x}}dx \\ &= -\dfrac{6}{7}\log_ex +c \end{align}

Integration of $\displaystyle \dfrac{f'(x)}{f(x)}$

\large \begin{align} \displaystyle \dfrac{d}{dx}\log_e{f(x)} &= \dfrac{1}{f(x)} \times f'(x) \\ &= \dfrac{f'(x)}{f(x)} \\ \log_e{f(x)} &= \int{\dfrac{f'(x)}{f(x)}}dx \\ \therefore \int{\dfrac{f'(x)}{f(x)}}dx &= \log_e{f(x)} +c \end{align}

Example 5

Find $\displaystyle \int{\dfrac{4}{4x-3}}dx$.

\begin{align} \displaystyle \int{\dfrac{4}{4x-3}}dx &= \int{\dfrac{(4x-3)’}{4x-3}}dx \\ &= \log_e(4x-3) + c \end{align}

Example 6

Find $\displaystyle \int{\dfrac{1}{2x+3}}dx$.

\begin{align} \displaystyle \int{\dfrac{1}{2x+3}}dx &= \int{\dfrac{1}{2} \times \dfrac{2}{2x+3}}dx \\ &= \dfrac{1}{2} \int{\dfrac{2}{2x+3}}dx \\ &= \dfrac{1}{2} \int{\dfrac{(2x+3)’}{2x+3}}dx \\ &= \dfrac{1}{2} \log_e(2x+3) +c \end{align}

Example 7

Find $\displaystyle \int{\dfrac{4}{5x+1}}dx$.

\begin{align} \displaystyle \int{\dfrac{4}{5x+1}}dx &= \dfrac{4}{5}\int{\dfrac{5}{5x+1}}dx \\ &= \dfrac{4}{5}\int{\dfrac{(5x+1)’}{5x+1}}dx \\ &= \dfrac{4}{5}\log_e(5x+1) +c \end{align}

Example 8

Find $\displaystyle \int{\dfrac{2x}{x^2+1}}dx$.

\begin{align} \displaystyle \int{\dfrac{2x}{x^2+1}}dx &= \int{\dfrac{(x^2+1)’}{x^2+1}}dx \\ &= \log_e(x^2+1) +c \end{align}

Example 9

Find $\displaystyle \int{\dfrac{x^2 + x + 1}{x}}dx$.

\begin{align} \displaystyle \int{\dfrac{x^2 + x + 1}{x}}dx &= \int{\Big(\dfrac{x^2}{x} + \dfrac{x}{x} + \dfrac{1}{x}\Big)}dx \\ &= \int{\Big(x + 1 + \dfrac{1}{x}\Big)}dx \\ &= \dfrac{x^2}{2} + x +\log_ex +c \end{align}

Note:
Many students made mistakes like the following:
\begin{align} \displaystyle \int{\dfrac{1}{x}}dx &= \int{x^{-1}}dx = \dfrac{x^{-1+1}}{-1+1}+c = \dfrac{x^{0}}{0}+c \end{align}
This is wrong and undefined, as its denominator is zero!
Please ensure $\displaystyle \int{\dfrac{1}{x}}dx=\log_ex + c$

Example 10

Find $\displaystyle \int{\dfrac{2x+1}{x+1}}dx$.

\begin{align} \displaystyle \dfrac{2x+1}{x+1} &= \dfrac{2x+2-1}{x+1} \\ &= \dfrac{2(x+1)}{x+1}-\dfrac{1}{x+1} \\ &= 2-\dfrac{1}{x+1} \\ \int{\dfrac{2x+1}{x+1}}dx &= \int{\Big(2-\dfrac{1}{x+1}\Big)}dx \\ &= 2x-\log_e{(x+1)} + c \end{align}

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