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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