1/[(X^2+1)(X+2)^2]
1/[X(X+1)^2]
please it is urgent
To learn, in general, how to work with partial fractions, try here. Then, since you haven't been able to start either of these, I'll give you the beginning:
The denominators are (x^2 + 1), (x + 2), and (x + 2)^2. So:
. . . . .$\displaystyle \frac{1}{(x^2\, +\, 1)(x\, +\, 2)^2}\, =\, \frac{Ax\, +\, B}{x^2\, +\, 1}\, +\, \frac{C}{x\, +\, 2}\, +\, \frac{D}{(x\, +\, 2)^2}$
Multiplying through gives:
. . . . .$\displaystyle 1\, =\, (Ax\, +\, B)(x\, +\, 2)^2\, +\, (C)(x^2\, +\, 1)(x\, +\, 2)\, +\, (D)(x^2\, +\, 1)$
Letting x = -2 gives:
. . . . .$\displaystyle 1\, =\, 0\, +\, 0\, +\, D(5)$
Use this to find the value of D. Then pick three other values for x to solve for the values of A, B, and C.
The factors in the denominators will be x, (x + 1), and (x + 1)^2. So use the standard set-up:
. . . . .$\displaystyle \frac{1}{x(x\, +\, 1)^2}\, =\, \frac{A}{x}\, +\, \frac{B}{x\, +\, 1}\, +\, \frac{C}{(x\, +\, 1)^2}$
Multiplying through gives:
. . . . .$\displaystyle 1\, =\, A(x\, +\, 1)^2\, +\, Bx(x\, +\, 1)\, +\, Cx$
Obvious values for x include x = 0 (which will give you the value of A) and x = -1 (which will give you the value of C).
If you get stuck, please reply showing how far you have gotten. Thank you!