How do you integrate #int(x+1)/((x+5)(x-1)(x-2))# using partial fractions?

Answer 1

#3/7ln|x-2| - 2/21ln|x+5| - 1/3ln|x-1| + c #

Since the factors on the denominator are linear , the the numerators of the partial fractions will be constants , say A , B and C.

#(x+1)/((x+5)(x-1)(x-2)) = A/(x+5) + B/(x-1) + C/(x-2) #

multiply through by (x+5)(x-1)(x-2)

#x+1 = A(x-1)(x-2)+B(x+5)(x-2)+C(x+5)(x-1) ...............(1)#

The aim now is to find the values of A,B and C. Note that if x=1 then the terms with A and C will be zero. If x =2 the terms with A and B will be zero and if x = -5 the terms with B and C will be zero. This is the starting point in finding A , B and C.

let x = 1 in (1): 2 = -6B # rArr B = -1/3#
let x = 2 in (1): 3 = 7C# rArr C = 3/7 #
let x = -5 in (1): -4 = 42A# rArr A = -2/21 #

Integral can now be written as :

#int(-2/21)/(x+5) dx -int (1/3)/(x-1) dx +int (3/7)/(x-2) dx #
#= 3/7ln|x-2| - 2/21ln|x+5| - 1/3ln|x-1| + c #
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Answer 2

To integrate (\frac{{x + 1}}{{(x + 5)(x - 1)(x - 2)}}) using partial fractions, you first decompose the rational function into partial fractions. The decomposition is as follows:

(\frac{{x + 1}}{{(x + 5)(x - 1)(x - 2)}} = \frac{A}{{x + 5}} + \frac{B}{{x - 1}} + \frac{C}{{x - 2}})

To solve for (A), (B), and (C), you can multiply both sides by the denominator ((x + 5)(x - 1)(x - 2)) and then equate coefficients of like terms. After finding the values of (A), (B), and (C), integrate each term separately.

Once you've integrated each partial fraction term, you will obtain the final result.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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