Integration technique

Evaluate





The radical term in the denominator is a hint for trigonometric substitution. However, this problem requires first a clever factoring manipulation and algebraic substitution before trigonometric substitution can be employed.

Let





Another substitution












From

the illustration below follows.









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Partial fractions

I already finished the contents of my workbook X and Why: Integration Techniques. I am now in the reproduction stage. While waiting for the finished product, let me present some of its contents.

Solve the following:
a)



b)



The problems look easy. Indeed, they are. So, how do we solve these?

For me, the basic step is to examine the function. There are several integration techniques but in these problems, it should be obvious that trigonometric integration, trigonometric substitution and integration by parts are not applicable. Why? The answer lies in the nature of the function in the given problems.

The power formula will not work because we cannot split these expressions to produce a function matching its derivative. Algebraic substitution will not work because the expressions are already simplified. The other option, which turns out to be applicable, is integration using partial fractions.

For a, note the like powers for numerator and denominator, so divide first. The first term in the answer should be obvious.



For b, there are non-repeating linear factors.





Easy.


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Reference: [4]

Death

Old mathematicians never die; they just lose some of their functions.

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source: [7]

Integration Techniques: Problems 2.1.27.23 and 2.1.29.23

Obtain the general solution for the following problems:

2.1.27.23]


2.1.29.23]

Obviously, trigonometric identities will come into play here. Problems like these will be covered in the integration techniques workshop.


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Reference: [5]

Mathematics

Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.

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Reference: [7]

Integration Techniques: Problem 3.17.21

Problem 3.17.21] Solve



This is one of the problems which looks intimidating because of its form. A simple technique can solve this although one approach is superior than the other.


We'll solve this in the workshop.

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Reference: [6]

Book: Fermat's Last Theorem

This is an immensely enjoyable treasure.

[image source: Amazon.com. 2009. Fermat's Last Theorem. http://ecx.images-amazon.com/images/I/51WX45QBQBL._SL500_AA240_.jpg. Last accessed: Oct 9, 2009.]

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Integration Techniques: Problem 3.14.20

Here's another problem which requires some integration.

Problem 3.14.20] Solve


Can you identify what technique will work for this kind of problem?


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Reference: [6]

Integration techniques: Problem 3.7.17.17

Problem 3.7.17.17] Solve

If you can solve this problem, maybe you should join me. I will offer a workshop on integration techniques soon. I am just finishing my workbook which contains:

1] Introduction. What basic weapons do you need to tackle integration problems? What areas will you find integration techniques useful?

2] Integration Techniques. By merely looking at the problem, how do you know which technique is applicable and will most likely work?

3] Problems. What are the various kinds of inegration problems? At least 10 problems will be given for each kind.

4] Solutions. Detailed solutions will be given. Alternative approaches will also be presented.

5] References.

My workbook is about 1/3 done. For now, I will post my selected problems.

You can join me in at least two capacities. First, you can serve as facilitator in workshops. Second, you can attend as a student. Third, you can donate funds.

The workshops will be held in series, ideally throughout the Philippines, wherever we are needed. This integration technique workshop is actually just one of the many activities I have in mind for the Lemniscate Engineering Institute. We will package programs and projects for this entity and hopefully, we can get sustained external funding.

In the past, we're just offering the Lemniscate Prize which is a monetary reward for undergraduate research in agricultural engineering and food technology. This is already on its 3rd year. This time, we're expanding. We'll build a library and we'll offer workshops. For the library, we already have about a dozen titles in mathematics, physics, agricultural engineering and food engineering.

We're doing this for many reasons. For one, I think we need more engineers and scientists than politicians.

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In search of Schrodinger's cat

I finished reading this last month. Nothing spectacular. Good enough for leisurely reading.

Separable differential equation

2.1.1.23] Solve




Solution:














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[Reference: 5]