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