We first expand the given expression into partial fractions:
(x+1)/((x+5)(x+6)(x-1)) -= A/(x+5)+B/(x+6)+C/(x-1)x+1(x+5)(x+6)(x−1)≡Ax+5+Bx+6+Cx−1
(x+1)/((x+5)(x+6)(x-1)) -= (A(x+6)(x-1) + B(x+5)(x-1) + C(x+5)(x+6))/((x+5)(x+6)(x-1))x+1(x+5)(x+6)(x−1)≡A(x+6)(x−1)+B(x+5)(x−1)+C(x+5)(x+6)(x+5)(x+6)(x−1)
And so. (x+1) -= A(x+6)(x-1) + B(x+5)(x-1) + C(x+5)(x+6))(x+1)≡A(x+6)(x−1)+B(x+5)(x−1)+C(x+5)(x+6))
Put x=-5=>-5+1=A(-5+6)(-5-1)+0+0x=−5⇒−5+1=A(−5+6)(−5−1)+0+0
:. (1)(-6)A=-4=>A=2/3
Put x=-6=>-6+1=0+B(-6+5)(-6-1)+0
:. (-1)(-7)A=-5=>B=-5/7
Put x=1=>1+1=0+0+C(1+5)(1+6)
:. (6)(7)C=2=>C=1/21
So the partial fraction decomposition is:
(x+1)/((x+5)(x+6)(x-1)) -= 2/(3(x+5))-5/(7(x+6))+1/(21(x-1))
We now want to integrate; so
int(x+1)/((x+5)(x+6)(x-1))dx = int(2/(3(x+5))-5/(7(x+6))+1/(21(x-1)))dx
int(x+1)/((x+5)(x+6)(x-1))dx = 2/3int1/(x+5)dx - 5/7 int1/(x+6)dx+1/21 int 1/(x-1)dx
int(x+1)/((x+5)(x+6)(x-1))dx = 2/3ln|x+5| - 5/7 ln|x+6|+1/21 ln|x-1| + c
