partial fraction decomposition

label Calculus
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Jun 4th, 2015

x^2 + 30x + 225 = (x+15)^2.

Make the substitution y=x+15, so x=y-15 and y is from 15 to 30. The indefinite integral is

int_[(y-15)^3/y^2] = int_[(y^3 - 3*15y^2 + 3*15^2*y - 15^3)/y^2] =

int_[y - 45 + 3*15^2/y - 15^3/y^2] =

(1/2)y^2 - 45y + 3*15^2*lny + 15^3/y.  Substitute y from 15 to 30 and obtain

(1/2)(30^2-15^2) - 45(30-15) + 3^15^2*(ln30-ln15) + 15^3*(1/30-1/15) =

(1/2)*15^2*3 - 15^2*3 + 15^2*ln2 - 15^3/30 =

(to be continued now)

Jun 4th, 2015

sorry, there is a mistake at the last string, 

= (1/2)*15^2*3 - 15^2*3 + 3*15^2*ln2 - 15^3/30 =

= 15^2*[3/2 - 3 + 3*ln2 - 1/2] = 225*[3*ln2 - 2].

this is can be written also as 675*ln2 - 450.

Jun 4th, 2015

awesome thanks a lot!!!

Jun 4th, 2015

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