How do you integrate #int(x+1)/((x+5)(x+3)(x+4))# using partial fractions?
The answer is
Let's perform the decomposition into partial fractions
The denominators are the same, we compare the numerators
So,
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To integrate ( \frac{{x+1}}{{(x+5)(x+3)(x+4)}} ) using partial fractions:

Factor the denominator: ( (x+5)(x+3)(x+4) ).

Write the partial fraction decomposition: ( \frac{{x+1}}{{(x+5)(x+3)(x+4)}} = \frac{{A}}{{x+5}} + \frac{{B}}{{x+3}} + \frac{{C}}{{x+4}} ).

Multiply both sides by the denominator ( (x+5)(x+3)(x+4) ) to clear the fractions.

Expand and equate coefficients to solve for ( A ), ( B ), and ( C ).

Once you have found ( A ), ( B ), and ( C ), integrate each term separately.

Finally, combine the integrals to get the result.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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