Integration By Partial Fractions
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- เผยแพร่เมื่อ 28 ก.ย. 2024
- This calculus video tutorial provides a basic introduction into integrating rational functions using the partial fraction decomposition method. Partial fraction decomposition is the process of breaking a single complex fraction into multiple simpler fractions. The integrals of many rational functions lead to a natural log function with absolute value expressions. This video explains what to do when you have repeated linear factors and quadratic factors. This tutorial contains many examples and practice problems on integration by partial fractions.
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I have a question: how do you know when to use U substitution, integration by parts, or partial fraction method when looking at an integral? What do you need to look out for in an integral to determine the method of solution? Basically what is a rule of thumb for these things?
Usually, you should try a u substitution as the first method. If that doesn’t work, then use integration by parts for multiplication or partial fractions for division. Eventually, you’ll be able to intuitively tell what integrals to use which techniques on, but it takes practice. I usually just do a u substitution in my head and then sus out whether the u substitution will take care of the entirety of the integral.
genuinely you cannot know for sure it just takes practice and working a lot on a problem while trying everything
As a general rule of thumb, I always start with u substitution.
In case it didn't work, I remind myself of several procedures based on the first glance at the problem:
if it contains a lone natural log, exponential function (especially e^x) or inverse trig function, I would try By Parts first.
if it contains powers of sec and tan, or sin and cos, I would try Identities method first.
if it contains sqrt(u^2 +- a^2), I would try Trigonometric Substitution.
if it has polynomials (long expression), I would try Partial Fraction.
Practice
Try this th-cam.com/video/M5MaGUO0JDs/w-d-xo.html by BlackPenRedPen
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What other teachers don't seem to make us understand, you put it into a flowing liquid and make us drink it with ease. I have never understood integration by partial fraction like this before.
docs.google.com/document/d/15xkx77jWOq3Bv6JDzrRzMNKaH5a7l5N2AMHP_QLBBq4/edit?usp=sharing
Refer here also. A better one
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for the first one we could use 1/x^2-2^2 formula
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I have a question for the last problem, you used the trig substitution, and I haven't learnt that yet so I used Integration by parts and got the answer in different function. Is this allowed or trig substitution is the only way to solve this?
Thanks in advance!
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Hey. The last problem is wrong. You have with the rule of the natural log, you have to multiply the values together not divide them because an addition sign ties them together
why cant x^2 + 4 be factored at 27:00?
x^2+4 is not a difference of squares (it is a sum of squares), so it cannot be factored like x^2-1 in the example. Also, x^2+4 has no real solutions if you were to set it equal to 0 and solve for x (because no real number squared will equal -4), which is a good indicator that it won't be factorable. Hope this helps!
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Is there a specific type of problem you are looking for help with?
At 30:00 minute interval, he stated that there is no value for x that could make x^2+4=0. That is true for the real numbers. However, substituting x=2i would give the following:
=(2i)^2+4
-4+4=0
This is true since i=sqrt(-1)
Thus squaring it give -1 and squaring 2 gives you 4.
Just saying this for those who had not known this before hand.
How you are giving the 3rd value as 0 because there is no such thing as 0. In problem( x^2+9)/(x^2-1)(x^2+4)
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