AntiderivativesDetermine the antiderivative of y=1/(x-1)(x+4).
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
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
giorgiana1976 | College Teacher | (Level 3) Valedictorian
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