A093integralandlogs


 * The antiderivative of 1/x **

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

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


 * Example 1 **

math . \qquad \text{Find } \; \displaystyle{ \int \dfrac{3}{x} \; dx } \; \text{ for } x > 0 \qquad. math

__**Solution:**__

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


 * Example 2 **

math . \qquad \text{Find } \; \displaystyle{ \int \dfrac{2}{3x-2} \; dx } \; \text{ for } x > \dfrac{2}{3} \qquad. math

__**Solution:**__

math . \qquad \textit{Manipulate fraction to get it in the form } \; \dfrac{g'(x)}{g(x)} \qquad. math

math . \qquad \displaystyle{ \int \dfrac{2}{3x-2} \; dx } \\. \\ . \qquad = 2 \displaystyle{ \int \dfrac{1}{3x-2} \; dx } \\. \\ . \qquad = \dfrac{2}{3} \displaystyle{ \int \dfrac{3}{3x-2} \; dx } \\. \\ . \qquad = \dfrac{2}{3} \log_e \big( 3x-2\big) + c, \quad 3x - 2 > 0 \qquad. \\ . \\ . \qquad = \dfrac{2}{3} \log_e \big( 3x-2\big) + c, \quad x > \dfrac{2}{3} math


 * Antiderivative into log e |x| **

(not in course, but you will not be marked wrong if you do this)

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


 * Note: ** If no domain is specified, then we should use the maximal domain which is R\{0}


 * Example 3 **

math . \qquad \text{Find } \; \displaystyle{ \int \dfrac{1}{5-x} \; dx } \qquad. math


 * Solution:**

math . \qquad \textit{Manipulate fraction to get it in the form } \; \dfrac{g'(x)}{g(x)} \qquad. math

math . \qquad \displaystyle{ \int \dfrac{1}{5-x} \; dx } \\. \\ . \qquad = - \displaystyle{ \int \dfrac{-1}{5-x} \; dx } \\. \\ . \qquad = - \log_e | 5 - x | + c, \quad 5 - x \neq 0 \qquad. \\ . \\ . \qquad = - \log_e |5 - x | + c, \quad x \neq 5 math

.