Problem

Power-log.png

Consider the family of functions

.

Part 1: Calculate the indefinite integral

.

Part 2: For what values does the following integral converge?

Solution

Part 1

This family of functions is the product of a power function with the natural logarithm function. Therefore the method of integration by parts should be used.

According to LIATE, let and .

Then and .

Therefore,

Part 2

From the result of the indefinite integral, it is obvious that . Just look at the denominators. Now this means

.

Notice that if , the power functions become reciprocal functions, which means the integral diverges when evaluating the lower bound. However if , the integral converges to the expression

.

Therefore the integral

converges if .

Community content is available under CC-BY-SA unless otherwise noted.