09 2antiderivln

# The antiderivative of 1/x

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Since $\dfrac{d}{dx} \Big( \; \log_e (x) \; \Big) = \dfrac{1}{x} \quad \text{ for } x > 0$
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We get $\displaystyle{ \int \dfrac{1}{x} \; dx } = \log_e (x) + c \quad \text{ for } x > 0$

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Since $\dfrac{d}{dx} \Big( \; \log_e \big( \, g(x) \, \big) \; \Big) = \dfrac{g'(x)}{g(x)} \quad \text{ for } g(x) > 0$
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We get $\displaystyle{ \int \dfrac{g'(x)}{g(x)} \; dx } = \log_e \big( \, g(x) \, \big) + c \quad \text{ for } g(x) > 0$

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### Example 1

Find $\displaystyle{ \int \dfrac{3}{x} \; dx } \quad \text{ for } x > 0$

Solution:

… … $\displaystyle{ \int \dfrac{3}{x} \; dx }$
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… … $= 3 \displaystyle{ \int \dfrac{1}{x} \; dx }$
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… … $= 3 \log_e (x) + c, \qquad x > 0$

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### Example 2

Find $\displaystyle{ \int \dfrac{2}{3x-2} \; dx } \quad \text{ for } x > \dfrac{2}{3}$

Solution:

… … Manipulate fraction to get it in the form: $\dfrac{g'(x)}{g(x)}$
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… … $\displaystyle{ \int \dfrac{2}{3x-2} \; dx }$
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… … $= 2 \displaystyle{ \int \dfrac{1}{3x-2} \; dx }$
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… … $= \dfrac{2}{3} \displaystyle{ \int \dfrac{3}{3x-2} \; dx }$
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… … $= \dfrac{2}{3} \log_e \big( 3x-2\big) + c, \quad 3x - 2 > 0$
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… … $= \dfrac{2}{3} \log_e \big( 3x-2\big) + c, \quad x > \dfrac{2}{3}$
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## Antiderivative into loge|x|

• This is in the Specialist Maths Course but not in Maths Methods course,
• You will not be marked wrong if you do this in Maths Methods

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Since $\dfrac{d}{dx} \Big( \; \log_e |x| \; \Big) = \dfrac{1}{x}, \quad \text{ for } x \in R \backslash \{ 0 \}$
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We get $\displaystyle{ \int \dfrac{1}{x} \; dx } = \log_e |x| + c, \quad \text{ for } x \in R \backslash \{ 0 \}$

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Note: If no domain is specified, then we should use the maximal domain which is $R \backslash \{ 0 \}$ in Specialist Maths

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### Example 3

Find $\displaystyle{ \int \dfrac{1}{5-x} \; dx }$

Solution:

… … Manipulate fraction to get it in the form $\dfrac{g'(x)}{g(x)}$
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… … $\displaystyle{ \int \dfrac{1}{5-x} \; dx }$
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… … $= - \displaystyle{ \int \dfrac{-1}{5-x} \; dx }$
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… … $= - \log_e \big| 5 - x \big| + c, \qquad 5 - x \neq 0$
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… … $= - \log_e \big|5 - x \big| + c, \qquad x \neq 5$
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