Discovery Project: Patterns in Integrals: Question 2: sin(ax)cos(bx)

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In this video I go over Question 2 of the Discovery Project: Patterns in Integrals section of my calculus book. In this question I go over the family of integrals of the form with integrand sin(ax)cos(bx) and show how we can derive the general integral formula by finding the pattern that arises from using a computer to quickly solve different examples within that integral family.


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Discovery Project: Patterns in Integrals: Question 2

Discovery Project Integral Patterns Question 2.jpeg

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 2:

(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 the techniques shown in my Trigonometric Integrals video series. For what values of a and b is it valid?

Solution:


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