**Problem**

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 .

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