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...
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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