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Integration of rational fraction

Integration of rational fraction:

In order to convert improper rational function into a proper one, we can use long division:

P(x)Q(x)=F(x)+R(x)Q(x),

where F(x) is a polynomial, R(x)Q(x) is a proper rational function.

To integrate a proper rational function, we can apply the method of partial fractions.

This method allows to turn the integral of a complicated rational function into the sum of integrals of simpler functions.

The denominators of the partial fractions can contain non-repeated linear factors, repeated linear factors, non-repeated irreducible quadratic factors, and repeated irreducible quadratic factors.

To evaluate integrals of partial fractions with linear or quadratic denominators, we use the following 6 formulas:

1.Adxax+b=Aln|ax+b|

2.Adx(ax+b)k=Aa(1k)(ax+b)k1

For partial fractions with irreducible quadratic denominators, we first complete the square:

ax2+bx+c=a[(x+b2a)2+4acb24a2].

Hence, we can write:

Ax+B(ax2+bx+c)kdx=At+B[a(t2+m2)]kdt=1akAt+B(t2+m2)kdt,

where

t=x+b2a,m2=4acb24a2,B=BAb2a.

The possible cases for fractions with quadratic denominators are covered by the following integrals:

3.tdtt2+m2=12 ln(t2+m2)

4.dtt2+m2=1mtan-1tm

5.tdt(t2+m2)k=12(1k)(t2+m2)k1

Finally, the integral dt(t2+m2)k can be evaluated in k steps using the reduction formula

6.dt(t2+m2)k=t2m2(k1)(t2+m2)k1+2k32m2(k1)dt(t2+m2)k1