How do you integrate #(4x^2-x+3)/((x+5)(x-1)(x-2))# using partial fractions?

Answer 1

#-ln|x-1|+17/7ln|x-2|+18/7ln|x+5|+K#, or,

#ln|((x+5)^(18/7)(x-2)^(17/7))/(x-1)|+K#.

Let #I=int(4x^2-x+3)/((x+5)(x-1)(x-2))dx#.

We will use the Method of Partial Fraction to split the Integrand

#(4x^2-x+3)/((x+5)(x-1)(x-2))=A/(x-1)+B/(x-2)+C/(x+5);#
where, #A,B,C in RR#.
To determine, #A,B,C#, let us use Heavyside's Cover-up Method :-
#A=[(4x^2-x+3)/((x+5)(x-2))]_(x=1)=6/-6=-1#;
#B=[(4x^2-x+3)/((x+5)(x-1))]_(x=2)=17/7#;
#C=[(4x^2-x+3)/((x-1)(x-2))]_(x=-5)=108/((-6)(-7))=18/7#.
Therefore, #I=int[-1/(x-1)+(17/7)/(x-2)+(18/7)/(x+5)]dx#
#=-ln|x-1|+17/7ln|x-2|+18/7ln|x+5|+K#, or,
#=ln|((x+5)^(18/7)(x-2)^(17/7))/(x-1)|+K#.

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Answer 2

To integrate the rational function (4x^2 - x + 3)/((x + 5)(x - 1)(x - 2)) using partial fractions, follow these steps:

  1. Perform polynomial long division if necessary to ensure the degree of the numerator is less than the degree of the denominator.
  2. Express the rational function as a sum of partial fractions.
  3. Find the constants for the partial fractions.
  4. Integrate each partial fraction separately.
  5. Combine the results to obtain the final integral.

If you need further clarification or assistance with any step, feel free to ask.

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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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