Evaluating the Integral: ∫ (log x)/x dx

Integrating functions involving logarithms can be a challenging task for many high school and college students. One such integral that often appears in mathematics curricula is the evaluation of ∫ (log x)/x dx.

Understanding the Integral

The integral ∫ (log x)/x dx represents the accumulated area under the curve of the function (log x)/x over a given interval. The logarithmic function, log x, appears in the numerator, while the variable x appears in the denominator.

Step-by-Step Evaluation

To evaluate this integral, we can use the technique of integration by parts. The steps are as follows:

1. Let u = log x and dv = 1/x dx.
2. Then, du = 1/x dx and v = -x.
3. Applying the integration by parts formula, we get: ∫ (log x)/x dx = -x log x + ∫ x^-1 dx
4. Recognizing that ∫ x^-1 dx = ln |x| + C, where C is the constant of integration, we can substitute this into the previous step: ∫ (log x)/x dx = -x log x + ln |x| + C

Evaluating the Integral over a Specific Interval

To evaluate the integral over a specific interval, say [a, b], we would substitute the upper and lower limits into the final expression:

∫a^b (log x)/x dx = [-x log x + ln |x|]a^b

This would give us the net area under the curve of (log x)/x between the points a and b on the x-axis.

Conclusion

The integral ∫ (log x)/x dx is a fundamental calculus problem that demonstrates the application of integration by parts. By understanding the step-by-step process, students can confidently evaluate this integral and apply the techniques to a variety of other logarithmic and rational functions.