7.4 Integration by Partial Fractions

Integration by Partial FractionsThe method of Partial Fractions is used to integraterational is, we want to compute P(x)Q(x)dxwhereP,Qare reduce1the integrand to the formS(x) +R(x)Q(x)where R< we write the integrand as a polynomial plus a rational function7x+2whose denom-inator has higher degreee than its numerator. Thankfully, this expression can be easily integratedusing +3x+2=x(x+2) 2x+3x+2=x+ 2(x+2) +4+3x+2=x 2+7x+2= x2+3x+2dx= x 2+7x+2dx=12x2 2x+7 ln|x+2|+cWhat if Q 2?If the denominatorQ(x)is quadratic or has higher degree, we need another that R< Q. Then the rational functionR(x)Q(x)can be written as a sum of Fractions of theformA(ax+b)mAx+B(ax2+bx+c)nwhere A,B,a,b,c are constants and m,n are positive such as the above can all be integrated using either logarithms or trigonometric a little experimenting, you should be convinced that3x2+2x+3x3+x=3x+21+x2It follows that 3x2+2x+3x3+xdx=3 ln|x|+2 tan 1x+cThe burning question ishowto find the expressions in the Therorem.

Since the denominators are equal, it follows that the numerators are equal: x +8 = A(x +2)+ B(x 1) This is a relationship between A, B which holds for all3 x: every value of x gives a valid rela-tionship between A and B. Evaluating at x = 1 and x = 2 gives two very simple expressions: x = 1 : 9 = 3A =)A = 3 x = 2 : 6 = 3B =)B = 2

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  Fractions, Partial, Integration, Denominators, 4 integration by partial fractions

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