How do you integrate #(6x+1)/(x^2+2x+3)#?
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To integrate the expression (6x + 1)/(x^2 + 2x + 3), you can use partial fraction decomposition. First, factor the denominator as (x + 1 + 2i)(x + 1 - 2i), where i is the imaginary unit. Then, express the fraction as the sum of two partial fractions:
(6x + 1)/(x^2 + 2x + 3) = A/(x + 1 + 2i) + B/(x + 1 - 2i)
Find A and B by multiplying both sides by the denominator of the fraction and then equating coefficients of like terms. After finding A and B, integrate each partial fraction separately with respect to x. The integral of A/(x + 1 + 2i) can be found using the natural logarithm, and the integral of B/(x + 1 - 2i) can also be found using the natural logarithm. Finally, add the integrals together to get the overall solution.
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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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