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 $\displaystyle \int{\dfrac{2}{x}}dx$.

Example 2

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

Example 3

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

Example 4

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

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


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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

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

Note:
Many students made mistake 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 it denominator is zero!
Please ensure $\displaystyle \int{\dfrac{1}{x}}dx=\log_ex + c$

Example 10

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






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