Natural Logarithmic Function and Integration

Natural Logarithmic Function and its Graph

The natural logarithm of a positive number x, written as ln x, is the value of an integral. The natural logarithmic function f(x) = ln x, with e as the base of the logarithm, is defined by $LaTeX: \ln x=\int_1^x\frac{1}{t}dt$ . The domain of the natural logarithmic function is the set of all positive real numbers. Geometrically, if x > 1, then ln x is the area under the continuous curve y=1/t from t = 1 to t = x. This graph reinforces the idea of the natural logarithmic function as the antiderivative given by the differential equation $LaTeX: \frac{dy}{dx}=\frac{1}{x}$ .

Log Rule for Integration

The general rule for differentiating a natural logarithmic function is equation image indicator $LaTeX: \frac{d}{du}\left(\ln u\right)=\frac{1}{u}\frac{du}{dx}$ . The final result in the following differentiation of a natural logarithmic function problem has important implications for determining a corresponding integration formula.

Find f'(x) if f(x) = ln |x|

equation image indicator $LaTeX: f(x) = \begin{cases} \ln x & if\: x > 0\\ \ln(-x) & if\: x<0\\ \end{cases} \\ f'(x) = \begin{cases} \frac{1}{x} & if\: x > 0\\ \frac{1}{-x}(-1)=\frac{1}{x} & if\: x<0\\ \end{cases}$

Thus, equation image indicator $LaTeX: f'\left(x\right)=\frac{1}{x}$ for all $LaTeX: x\ne0$ AND $LaTeX: \frac{d}{dx}\left(\ln\left|x\right|\right)=\frac{1}{x}$

The related integral formula (Log Rule for Integration) states that if u is a positive or negative differentiable function, then equation image indicator $LaTeX: \int\frac{1}{u}du=\ln\left|u\right|+C$ . Recall $LaTeX: \int u^ndx=\frac{u^{n+1}}{n+1}+C,\:n\ne-1$ . The Log Rule for Integration provides the means for addressing the n ≠ -1 restriction.

Integration Involving Change of Variables

The integration technique using a change of variables followed by a substitution is valid for integrands involving natural logarithmic functions.

Recall that if u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then du = g'(x)dx, and equation image indicator $LaTeX: \int f\left(g\left(x\right)\right)g'\left(x\right)dx=\int f\left(u\right)du=F\left(u\right)+C$ .

View the video below from 3:35 - 7:46.

The examples that follow illustrate a variety of function types that require the log rule for integration.

Example 1

Evaluate the integral: $LaTeX: \int\frac{\cos\:x}{2+\sin x}dx$ text annotation indicator

Solution: Let $LaTeX: u=2+\sin x,\:du=\cos xdx$

$LaTeX: \int\frac{\cos x}{2+\sin x}dx=\int\frac{du}{u}=\ln\left|u\right|+C=\ln\left|2+\sin x\right|+C$

Example 2

Evaluate the integral: $LaTeX: \int \frac{2-x^2}{6x-x^3}dx$ text annotation indicator

Solution: Let $LaTeX: u=6x-x^3;\:du=\left(6-3x^2\right)dx=3\left(2-x^2\right)dx$

$LaTeX: \int\frac{2-x^2}{6x-x^3}dx=\frac{1}{3}\int\frac{du}{u}=\frac{1}{3}\ln\left|u\right|+C=\frac{1}{3}\ln\left|6x-x^3\right|+C$

Example 3

Evaluate the integral: $LaTeX: \int\frac{e^x}{e^x+1}dx$ text annotation indicator

Solution: $LaTeX: u=e^x+1,\:du=e^xdx$

$LaTeX: \int\frac{e^x}{e^x+1}dx=\int\frac{du}{u}=\ln\left|u\right|+C=\ln\left|e^x+1\right|+C$

Example 4

Evaluate the integral: $LaTeX: \int_1^2\frac{4+u^2}{u^3}du$ text annotation indicator

Solution: $LaTeX: =\int_1^2\frac{4}{u^3}du+\int_1^2\frac{u^2}{u^3}du\\ =4\int_1^2u^{-3}du+\int_1^2\frac{1}{u}du\\ =4\left(\frac{1}{-2u^2}\right)\Bigg]_1^2+\ln\left|u\right|\Bigg]_1^2\\ =\left(\frac{-2}{u^2}\right)\Bigg]_1^2+\ln\left|u\right|\Bigg]_1^2\\ =\left(\frac{-1}{2}+2\right)+\ln2-\ln1=\frac{3}{2}+\ln2\\$

Integration Involving Long Division

Recall the technique for finding end behavior or slant asymptotes arising when graphing a rational function equation image indicator $LaTeX: h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ , where the degree of the numerator is greater than the degree of the denominator. q(x) and r(x) are the quotient and remainder when f(x) is divided by g(x), i.e., $LaTeX: h\left(x\right)=q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$ with the degree of r less than the degree of g. The graph of q is the end behavior asymptote of h. Similarly, when integrals involve a rational function equation image indicator $LaTeX: h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ , where the degree of the numerator is greater than or equal to the degree of the denominator, performing long division and then rewriting the integrand as $LaTeX: q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$ with the degree of r less than the degree of g transforms the equation image indicator $LaTeX: \frac{r\left(x\right)}{g\left(x\right)}$ portion of the integral into an integrable form $LaTeX: \int\frac{1}{u}du=\ln\left|u\right|+C$ .

Integrals of Trigonometric Functions

The Log Rule for Integration is used to develop formulas for integrating four trigonometric functions that were previously not possible. View the presentation on deriving the integrals of tan x, sec x, and csc x.

Integration rules for trigonometric functions are summarized below.

The fnInt calculator command is featured in the next presentation beginning at 7:30.

Natural Logarithmic Function and Integration Practice

Natural Logarithmic Function and Integration: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the natural logarithmic function and integration.

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