Lecture 15 Partial Fraction Decomposition - W15 MAT 21B...

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W15 MAT 21B LECTURE 15: PARTIAL FRACTION DECOMPOSITION GEORGE MOSSESSIAN A rational function is just a function which is a quotient of two polynomials, like 5 x - 3 x 2 - 2 x - 3 , for example. Partial fraction decomposition is the technique of breaking this kind of function up into pieces which we can individually integrate. For example, we’ll figure out how to rewrite that fraction as 2 x + 1 + 3 x - 3 , which is easy to integrate to get 2 ln | x 1 | + 3 ln | x - 3 | + C . Example 1. You may have noticed that the denomenator of that function factors as ( x +1)( x - 3), which is why we split the fraction between those two denomenators – the common denomenator is exactly x 2 - 2 x - 3. To find the numerators of the decomposed sum, we write 5 x - 3 x 2 - 2 x - 3 = A x + 1 + B x - 3 = B ( x + 1) + A ( x - 3) = x ( B + A ) + B - 3 A = 5 x - 3 which gives us the system of equations ( A + B = 5 - 3 A + B = - 3 , which has the solution A = 2, B = 3. Example 2. If the numerator has a higher power than the denomenator, we can use long division to get a remainder, which we then use partial fraction decomposition on.

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