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

Answer 1

#-1/4 ln(x+1) -ln (x-2) +5/4 ln (x-3) +c#

Before integration, partial fractions can be done as explained below

The partial fractions would thus be

#-1/(4(x+1)) -1/(x-2)+5/(4(x-3)#

Integration is now simple #-1/4 int dx/(x+1) -int dx/(x-2) +5/4 int dx/(x-3)#

=#-1/4 ln(x+1) -ln (x-2) +5/4 ln (x-3) +c#

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Answer 2

To integrate ( \frac{2x - 1}{(x + 1)(x - 2)(x + 3)} ) using partial fractions, follow these steps:

  1. First, express the rational function as the sum of partial fractions: [ \frac{2x - 1}{(x + 1)(x - 2)(x + 3)} = \frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{x + 3} ]

  2. Multiply both sides by the denominator ( (x + 1)(x - 2)(x + 3) ) to clear the fractions: [ 2x - 1 = A(x - 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x - 2) ]

  3. Expand and equate coefficients of like terms. This will give you a system of linear equations to solve for ( A ), ( B ), and ( C ).

  4. Once you find the values of ( A ), ( B ), and ( C ), rewrite the original function using these partial fractions.

  5. Integrate each partial fraction separately.

  6. Finally, add the integrated partial fractions together to 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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