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Rules of Integration for Indefinite Integral
  Method of Partial Fractions for Indefinite Integral
   Integration by Partial Fractions

Rules of Integration for Indefinite Integral

Method of Partial Fractions for Indefinite Integral

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where Φ(x) and φ(x) are rational, integral, algebraical functions of x

The integrand after denominator factorization is

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Integration by Partial Fractions

The method of integration by partial fractions can be expressed as

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The integral of polynomial Q(x) can be obtained by making use of the constant multiple, sum of function properties of intergration and applying the anti-differentiation of the derivative of polynomial. The integration of the four case of fraction factors after partial fractions are:

  1. linear partial fraction A/(ax+b)

    This is the most common type of partial fraction, the integral of the linear partial fraction can be obtained by the quotient of standard functions rule of indefinite integral,

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    Therefore the integral of linear partial fraction can be determined by:

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  2. repeated linear partial fraction  A/(ax+b)k

    This is the most simple type of partial fraction.The integral of the repeated linear partial fraction can be obtained by product of standard functions rule of indefinite integral.

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    Therefore the integral of repeated linear partial fraction can be determined by:

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  3. quadratic partial fractionr (Ax+B)/(ax2+bx+c)

    For the quadratic partial fraction, there are some variant forms.

    1. Standard quadratic partial fraction (Ax+B)/(ax2+bx+c).

      1. If the numerator can be expressed as the derivative of the denumerator, imply

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      2. If the numerator can not be expressed as the derivative of the denumerator, imply

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        For the last integral, the quadratic factor can be resolved by completing the square, imply

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        Since the quadratic factor is irreducible, 4ac-b2>0, imply

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        Therefore the integral is

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    2. Quadratic partial fraction Ax/(ax2+bx+c). Same as a.ii case where B=0, imply

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    3. Quadratic partial fraction B/(ax2+bx+c). Same as the last integral in a.ii case where B=D, imply

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    4. Quadratic partial fraction with quadratic factor, (ax2+bx)

      Quadratic factor (ax2+bx) is same as (ax2+bx+c) by letting c=0, the integral can be obtained as in case a

    5. Quadratic partial fraction with quadratic factor, (ax2+c)

      Quadratic factor (ax2+c) is same as (ax2+bx+c) by letting b=0, the integral can be obtained as in case a. But no complete the square is needed since the factor can be directly transformed into needed format., imply

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  4. repeated quadratic factor  (Ax+B)/(ax2+bx+c)k

    The proceduce of determine the integral is same as case 3 except for the determining of the integral with power after decomposing the original integral into two integrals. Imply

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    The first integral can be obtained by simple substitution. Imply

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    The second integral can be obtained by simple substitution. Imply

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    The second integral can further be expanded using the formule in integration by part as:

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    The theta angle can be transformed back to x through trigonometry, imply:

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    And the integral can be expressed as:

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ID: 111000033 Last Updated: 10/30/2011 Revision: 0 Ref:

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References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
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