How do you integrate #int (x+4)/(x^2 + 2x + 5)# using partial fractions?
# int \ (x+4)/(x^2+2x+5) \ dx = 1/2 \ ln |x^2+2x+5| + 3/2 \ arctan((x+1)/2) + C #
We seek:
Since we know that the denominator does not factorize with real factors, we can decompose the integrand as follows instead of using partial fractions:
When the numerator has been altered to become the denominator's derivative, this means that:
After that, substituting results in (without the integration constant):
After making the substitution again, we obtain:
Additionally, we are able to substitute, Let:
And by substituting, we obtain (without the integration constant):
After making the substitution again, we obtain:
After combining the two results and adding the integration constant, we obtain:
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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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