INTEGRALS REPORT
4-step-strategy Group 5
introduction The integral is one of the most important fields of Mathematics. They can be used in either conceptual problems, such as computing the areas between 2 curves; or realistic problems such as figuring out the hydrostatic pressure in deep-sea, finding the center of mass of an irregular object or calculating the volume of blood across a cross-section within a time interval. However, integral is not as straightforward as derivative. There are no rules that absolutely guarantee obtaining an integral of a function. Therefore, in this report we discuss a strategy for integration, which is the 4-step-strategy mentioned in Chapter 7.5 of out textbook.
Team members Huynh Viet Thong (C) - 13ECE2 Tran Hung Tri - 13ECE2 Nguyen Hoang Nhat - 13ECE2 Tran Dai Ngoc Hai - 13ECE2 Nguyen Khac The - 13ECE2
About the strategy The 4-step-strategy includes: 1. Simplify: The fundamental step to make the integral become clear and easy to find the way to solve. 2. Look for a substitution to solve the problem: If there is no obvious substitution, we can skip this step. 3. Classify: If we cannot find the solution after step 1 and step 2, we should try to define the form of the integral such as trigonometric, rational, integration by parts... 4. Try again: Try substitution, try integration by parts and try manipulating. Personally, this strategy is very helpful in computing the integrals: • Firstly, in the first step, we can split complicated integrals into easier forms that enable us to find out the correct answer. • Furthermore, we can use from the simplest to the most complicated method such as from substitution to radicals respectively, which create a mindset of solving integrals. • Finally yet importantly, the last step (Try Again) teach you to become more patient when solving the integrals because the integrals is not always easy to be solved. If we not apply this method, you may become confusing and cannot solve problems correctly. Therefore, we will deliver 2 examples to illustrate the usefulness of this strategy.
example 1
Without applying the 4-step-strategy, it is very hard to come up with the final result of the integral since it is quite complicated. However, we are able to solve this integral with the strategy since we try manipulating and simplifying after trying again in the 4th steps.
example 2
The preceding examples demonstrate strategy for evaluating integrals. In these examples, the guidelines 4-step are very helpful to follow as we may need to use identities, integration by part, and occasionally a little ingenuity.
about the CAS
All the integrals we have learned up to now are elementary integrals, which is mentioned in the textbook. When the problem is one of these cases, we can obtain the final result by using the method of substitution, integration by parts or formula that is unique to these cases, which is mentioned in the sections from 7.1 to 7.4. However, when there is a problem that is out of these cases, it becomes unsolvable since it is out of our knowledge at the moment. Let’s take one example of this case: This kind of anti-derivative does not belong to any kind of equation we have learned so far. Note that, this is NOT exponential function. Therefore, we cannot solve this problem. However, by using the CAS (Computer Algebra System), we can still receive the correct result. The point is, the CAS can solve any integrals, independent from its kind. However, can we understand what is written in the result? Turning back to the example of finding the anti-derivative of . By using CAS, the final result is: in which the erfi (x) is “imaginary error function�, which we have not studied so far. Therefore, despite the fact that CAS can solve any integrals, it is still crucial to know what we are doing with the CAS.
Summary In this report, we present the 4-step-strategy, which is crucial when solving integrals. We come to this final conclusion after analyzing each step and taking 2 examples to illustrate this point of view. In additional, we can use the CAS to calculate integrals, but it is obvious that we don’t understand what is written in the result in most cases since it is out of our understanding at the moment.
Main results After studying this report, the main points we archive is how can we apply the 4-step-strategy of solving integrals. We become familiar with the strategy, from simplifying to applying different methods. Sometimes, the integrals is very complicated and need the CAS to be calculated. Finally yet importantly, we are aware that we should be patient when computing integrals, try again and again if we cannot solve it from the first time.
Reference • Calculus Early Transcendentals 5th Edition - James Stewart • Calculus Concepts and Contexts 2nd Edittion - James Stewart & Brooks Cole