# How do you integrate #[(x + 4)/(x^2 + 8 x +17)] dx#?

Use substitution:

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To integrate (\frac{{x + 4}}{{x^2 + 8x + 17}}) with respect to (x), you can use partial fraction decomposition. First, factor the denominator (x^2 + 8x + 17) into ((x + 4)(x + 4 + \sqrt{3}i)), where (\sqrt{3}i) is the imaginary unit.

The partial fraction decomposition of (\frac{{x + 4}}{{x^2 + 8x + 17}}) is: [\frac{{x + 4}}{{x^2 + 8x + 17}} = \frac{{A}}{{x + 4}} + \frac{{B}}{{(x + 4 + \sqrt{3}i)}} + \frac{{C}}{{(x + 4 - \sqrt{3}i)}}]

To find (A), (B), and (C), multiply both sides by the denominator (x^2 + 8x + 17) and then equate the numerators. After solving for (A), (B), and (C), you integrate each term separately.

The integral of (\frac{{A}}{{x + 4}}) is (A\ln|x + 4|), the integral of (\frac{{B}}{{x + 4 + \sqrt{3}i}}) is (B\ln|x + 4 + \sqrt{3}i|), and the integral of (\frac{{C}}{{x + 4 - \sqrt{3}i}}) is (C\ln|x + 4 - \sqrt{3}i|).

Finally, you can combine these results to get the overall integral.

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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.

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