602LogExamples


 * Examples of Integrating into Natural Logarithms **

We have: math \displaystyle{\int} \dfrac{1}{x} dx = \log_e |x| + c, \; \; \text { where } x \neq 0 \qquad. math and math \displaystyle{\int} \dfrac{f'(x)}{f(x)} dx = \log_e |f(x)| + c, \; \; \text { where } f(x) \neq 0 \qquad. math

math \displaystyle{\int} \dfrac{5}{2x+3} dx \qquad. math
 * Example 1 **

Adjust numerator so that it is the derivative of the denominator (2) math \\ \displaystyle{\int} \dfrac{5}{2x+3} dx = 5\displaystyle{\int} \dfrac{1}{2x+3} dx \qquad. \\ . \\ . \qquad \qquad \quad = \dfrac{5}{2} \displaystyle{\int} \dfrac{2}{2x+3} dx \qquad. \\ . \\ . \qquad \qquad \quad = \frac{5}{2} \log_e |2x+3| + c, \; \; x \neq -\frac {3}{2} \qquad. math

math \displaystyle{\int} \dfrac{6x+5}{x^2} dx \qquad. math
 * Example 2 **

Separate into two fractions, simplify, then integrate math \\ \displaystyle{\int} \dfrac{6x+5}{x^2} dx = \displaystyle{\int} \dfrac{6x}{x^2} +\dfrac {5}{x^2} dx \qquad. \\ . \\ . \qquad \qquad \quad = \displaystyle{\int} \dfrac{6}{x} +\dfrac {5}{x^2} dx \qquad. math

math \\ . \qquad \qquad \quad = \displaystyle{\int} \dfrac{6}{x} +5x^{-2} dx \qquad. \\ . \\ . \qquad \qquad \quad = 6\log_e |x| - 5x^{-1} + c \qquad .\\. \\ . \qquad \qquad \quad = 6\log_e |x| - \dfrac {5}{x} + c, \; \; x \neq 0 \qquad. math

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