Determine the antiderivative of y=1/(x-1)(x+4).

To determine the antiderivative of y =
1/(x-1)(x+4).

To determine th
antiderivative.

We first split 1/(x-1)(x+4)  into partial fractions
.

1/((x-1)(x+4) = A/(x-1)+B/(x+4)

We
multiply both sides by (x-1)(x+4).

1 =
A(x+4)+B(x-1)....(1)

Put x= 1.Then 1 =
A(1+4)+B*0.

1=5A. So A = 1/5.

Purt x=
-4 in (1) : Then 1 = A*0 +B(-4 -1) = -5B.

Therefore 1 = -5B. So B  =
-1/5.

Therefore  1/(x-1)(x+1) = 1/5(x-1)
-1/5(x+1).

We now find the antiderivative of
1/(x-1)(x+4).

Int dx/(x-1)(x+1) = Int
{dx/5(x-1)-dx/5(x+1)}+C.

Int dx/(x-1)(x+1) = Int dx/5(x-1) - Int
dx/(5(x+1)+C.

Int dx/(x-1)(x+1) = (1/5) log (x-1) - (1/5) log
(x+1)+C.

Int dx/(x-10(x+1) = (1/5) { log(x-1) -
og(x+1)}+C.

Int dx/(x-1)(x+1) = (1/5)log
(x-1)/(x+1)+C

Therefore, the antiderivative of dx/(x-1)(x+1) is
(1/5)log (x-1)/(x+1)+C.