How do you integrate #int (x+5)/((x+3)(x-2)(x-7)) # using partial fractions?

Answer 1

The answer is #=1/25ln(|x+3|)-7/25ln(|x-2|)+6/25ln(|x-7|)+C#

Perform the decomposition into partial fractions

#(x+5)/((x+3)(x-2)(x-7))=A/(x+3)+B/(x-2)+C/(x-7)#
#=(A(x-2)(x-7)+B(x+3)(x-7)+C(x+3)(x-2))/((x+3)(x-2)(x-7))#

The denominators are the same, compare the numerators

#(x+5)=(A(x-2)(x-7)+B(x+3)(x-7)+C(x+3)(x-2))#
Let #x=-3#, #=>#, #2=50A#, #A=1/25#
Let #x=2#, #=>#, #7=-25B#, #B=-7/25#
Let #x=7#, #=>#, #12=50C#, #C=6/25#

Therefore,

#(x+5)/((x+3)(x-2)(x-7))=(1/25)/(x+3)+(-7/25)/(x-2)+(6/25)/(x-7)#

So, the integral is

#int((x+5)dx)/((x+3)(x-2)(x-7))=int(1/25dx)/(x+3)+int(-7/25dx)/(x-2)+int(6/25dx)/(x-7)#
#=1/25ln(|x+3|)-7/25ln(|x-2|)+6/25ln(|x-7|)+C#
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Answer 2

To integrate ( \frac{x + 5}{(x + 3)(x - 2)(x - 7)} ) using partial fractions, follow these steps:

  1. Factor the denominator: ( (x + 3)(x - 2)(x - 7) ).
  2. Write the fraction as a sum of partial fractions with undetermined coefficients.
  3. Determine the values of the unknown coefficients by equating the original expression to the partial fraction form.
  4. Integrate each partial fraction separately.
  5. Combine the results to find the final integrated expression.

The partial fraction decomposition will look like this:

[ \frac{x + 5}{(x + 3)(x - 2)(x - 7)} = \frac{A}{x + 3} + \frac{B}{x - 2} + \frac{C}{x - 7} ]

After finding the values of ( A ), ( B ), and ( C ), integrate each term separately.

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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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