How do you integrate #int (x+5)/((x+3)(x-2)(x-7)) # using partial fractions?
The answer is
Perform the decomposition into partial fractions
The denominators are the same, compare the numerators
Therefore,
So, the integral is
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To integrate ( \frac{x + 5}{(x + 3)(x - 2)(x - 7)} ) using partial fractions, follow these steps:
- Factor the denominator: ( (x + 3)(x - 2)(x - 7) ).
- Write the fraction as a sum of partial fractions with undetermined coefficients.
- Determine the values of the unknown coefficients by equating the original expression to the partial fraction form.
- Integrate each partial fraction separately.
- 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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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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