In this video I go over Question 3 of the Patterns in Integrals Discovery Project which explores how using a Computer Algebra System (CAS) to solve and find patterns of families of integrals can be used to derive integral formulas. In this question, I go over the family of integrals of the form xⁿ*ln(x) where n is not equal to -1. I derive the formula for the integral by guessing using the pattern in the integrals and prove that it is actually the case with the method of integration by parts.

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Discovery Project: Patterns in Integrals: Question 3

This is part of a section in my calculus book called "Discovery Project" which explores specific topics in calculus in greater detail.

In this project a computer algebra system (CAS) is used to investigate indefinite integrals of families of functions. By observing the patterns that occur in the integrals of several members of the family, we will be able to first guess, and then prove, a general formula for the integral of any member of the family.

Question 3:

(a) Use a computer algebra system to evaluate the following integrals:

(b) Based on the pattern of your responses in part (a), guess the value of the integral:

(c) Check your guess with a CAS. Then prove it using integration by parts. For what values of n is it valid?

Solution:

(a)

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