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

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The antiderivative is the indefinite integral.

We'll decompose the function into partial fractions:

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

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

We'll remove the brackets:

1 = Ax + 4A + Bx - B

We'll combine like terms:

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

A + B = 0

A = -B

4A - B = 1

4A + A = 1

5A = 1

A = 1/5

B = -1/5

1/(x-1)(x+4) = 1/5(x-1) - 1/5(x+4)

Int dx/(x-1)(x+4) = Int dx/5(x-1) - Int dx/5(x+4)

Int dx/(x-1)(x+4) = (1/5) (ln |x-1| - ln|x+4|) + C

The requested antiderivative is: Int dx/(x-1)(x+4) = (1/5)ln |(x-1)/(x+4)| + C

We have to find the antiderivative of y=1/(x-1)(x+4)

First let's express y as partial fractions.

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

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

=> Ax + 4A + Bx - B = 1

=> A + B = 0 and 4A - B = 1

Add the two,

5A = 1

A = 1/5

B = -1/5

y = 1/5(x - 1) - 1/5(x + 4)

Int [ y dy] = Int [ 1/5(x - 1) - 1/5(x + 4) dy]

=> (ln|x - 1| - ln|x + 4|)/5

=> ln[(x - 1)/(x + 4)]/5 + C

**The required derivative is ln[(x - 1)/(x + 4)]/5 + C**

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