Patterns in Integrals



Lab 2: Patterns in IntegralsSeattle Pacific University, MAT 1235, Calculus II ObjectivesTo use SageMath to explore the antiderivatives of some families of functions and to recognize patterns in the forms of these antiderivatives.To get practice with drawing conclusions from patterns of answers.Due Date: See CanvasDo not wait until the last minute to finish the lab. You never know what technical problems you may encounter (no papers in the printer, electricity is out, your best friend call and talk for 4 hours, alien attack…etc).Name 1:Name 2:Date:SAGE Commands: Indefinite Integrals1. Suppose we want to evaluate the integral . The SageMath comment is:show(integrate(1/x,x)); Note that the integration constant as well as the absolute value within the natural log function are not shown in the SageMath result, you need to put it back when typing your answers.2. The exponential function is represented in SageMath asexp(x);Lab ExercisesIn this lab you will use a computer to calculate antiderivatives of several functions. Each exercise concerns a family of related functions. You are to develop an intuition about the form of the antiderivatives of these families.In order to explain some of your answers, you may need to do additional SageMath experiments. Make sure you state the results clearly in your report. 1. a. Use SageMath to evaluate the following integrals of rational functions. Indefinite IntegralsAnswers b. (i) Based on the pattern of your answers in part (a), make a conjecture about the value of the indefinite integral where a and b are constants and . 1x+a(x+b)dx=(ii) Does your formula work when a or b is negative? Explain/provide evidence. (iii) Do a and b need to be integers for the formula to work? Explain/provide evidence. (iv) Does your formula work when a = b? Explain/provide evidence. If there are any exceptions, can you guess a separate formula that will work in those cases? 2. a. Use SageMath to evaluate the following integrals of powers of x times the natural logarithm.Indefinite IntegralsAnswers b. (i) Based on the pattern of your answers in part (a), make a conjecture about the value of the integral where n is a constant. xnlnx dx=(ii) Is the formula valid if n = 0? Explain/provide evidence. (iii) Does it work for all negative values of n? Explain/provide evidence. (iv) Does it work if n is not an integer? Explain/provide evidence. If there are any exceptions, can you guess a separate formula that will work in those cases? 3. a. Use SageMath to evaluate the following integrals of powers of x times the exponential function.Indefinite IntegralsAnswers b. Based on you answers above, make a conjecture about the form of the antiderivative of when n is a positive integer. When giving your answer, do not worry about the exact values of the constants involved, just give the general form of the functions.For example, if you think the integral of is a degree n polynomial, you writewhere are constants. xnex dx= ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download