In this video I go over another Question in the Patterns in Integrals Discovery Project and this time go over the family of integrals with integrand xⁿ * e^x where n is a positive integer. Like in the other questions, I use a Computer Algebra System (CAS) to solve multiple integrals with that form and based on the pattern of the answers guess a formula for the general form of the integral. I also use a CAS to calculate the general form but that answer involves the Gamma function, Γ(x) = (x-1)!, but that topic is beyond what I have covered thus far so it is shown as reference only in the video. I also prove the guess made from the pattern of integral answers using the Principle of Mathematical Induction. This is an important video to understand the power of computers in determining integral formulas for entire families of integrals as well as understanding mathematical induction so make sure to watch the video!

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Discovery Project: Patterns in Integrals: Question 4

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 4:

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 answer using a CAS
d) Based on the patterns in parts (a) and (b), guess the value of the integral:

where: n is a positive integer

e) Check your guess with a CAS. Then prove it using mathematical induction

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Solution:

a)

c)

e)

Note that Γ is called the Gamma function and it is basically an extension of the factorial function, !, but with its "argument" shifted down by 1.

I will go over the gamma function in later videos so for now, the answer from the CAS will be used just as a reference.

Proof using Mathematical Induction:

Recall from my earlier video about the Principle of Mathematical Induction:

• Let Sn be a statement about the positive integer n
• Suppose that:

1. S₁ is true.
2. S_k+1 is true whenever S_k is true.

• Thus S_n is true for all positive integers n.

In other words, if assuming that S_k is true results in showing that S_k+1 is true, then S_k = S_n is also true.

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