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قديم 11-14-2011, 04:06 PM   #1
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تاريخ التسجيل: Oct 2011
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افتراضي شرح وافي جدا لمحاضرة Partial Fractions

[align=center]أولا الشرح بالكلام طبعا بالانجليزي ولكن انجليزي سهل


The method of Partial Fractions is an extremely useful tool whenever you need to integrate a fraction with polynomials in both the numerator and denominator; something like this:


If you were asked to integrate

you shouldn’t have too much trouble, because if you don’t have a variable in the numerator of your fraction, then your integral is simply the numerator multiplied by the natural log (ln) of the absolute value of the denominator, like this:

where C is the constant of integration. Not too hard, right?
Don’t forget to use Chain Rule and divide by the derivative of your denominator. In the case above, the derivatives of both of our denominators are 1, so this step didn’t appear.
But if your integral is

then your answer will look like this:

The derivative of our denominator is 2, so we have to divide by 2, according to Chain Rule. Anyway, back to our original example. We said at the beginning of this section that

would be difficult to integrate, but that we wouldn’t have as much of a problem with

In fact, these two are actually the same function. If we try adding 3/(x+1) and 4/(x-1) together, you’ll see that we get back our original function.

Again, attempting to integrate

is extremely difficult. But if you can express this function as

then integrating is much simpler. This method of converting complicated fractions into simpler fractions that are easier to integrate is called decomposition into “partial fractions”.




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الشرح بالفيديوهات




شرح لمثال للتحليل الجزيئي ( Partial fraction )




في حالة المقام معادلة خطية يعني مافيهاش أسس تربيعية أو تكعيبية أو أكتر كلها أس واحد




مثال 1




http://vimeo.com/15702300



مثال 2



http://vimeo.com/15721842



مثال 3




http://www.youtube.com/watch?v=vEY4K4cw_BM





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دلوقت أمصلة على ان المقام برضو معادلة خطية ولكن المقام مكرر




المثال




http://vimeo.com/32019673





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هنا بقه مثال للمقام معادلة تربيعية وغير مكرر




http://vimeo.com/27490239



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المقام معادلة تربيعية ولكن مكرر




http://vimeo.com/27589137



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هنا بقه مثال يوضح لو ان درجة البسط طلعت أكبر من درجة المقام وبنضر للقسمة المطولة




ده مثال اهه سهل وبسيط نفهم منه القسمة المطولة











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وأخيرا لو انت ذاكرت كل ده او مذاكره هنا ملخص شامل للدرس كله يعني ممكن تطبعه في الآخر او تقراه كمراجعة بشرط انك تكون فهمت اللي فوق الأول





In summary, in order to integrate by expressing rational functions (fractions) in terms of their partial fractions decomposition, you should follow these steps:

1. Ensure that the rational function is “proper”, such that the degree (greatest exponent) of the numerator is less than the degree of the denominator. If necessary, use long division to make it proper.
2. Perform the partial fractions decomposition by factoring the denominator, which will always be expressible as the product of either linear or quadratic factors, some of which may be repeated.
a. If the denominator is a product of distinct linear factors: This is the simplest kind of partial fractions decomposition. Nothing fancy here.
b. If the denominator is a product of linear factors, some of which are repeated: Remember to include factors of lesser degree than your repeated factors.
c. If the denominator is a product of distinct quadratic factors: You must use the following two equations:

d. If the denominator is a product of quadratic factors, some of which are repeated: Use the two formulas above and remember to include factors of lesser degree than your repeated factors.












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سلامووو عليكم




بالتوفيق يا شباب[/align]
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