Consider the family of functions
Part 1: Calculate the indefinite integral
Part 2: For what values does the following integral converge?
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 .
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 .