# How to solve this integral #int_(-oo)^(+oo)(x)/((x^2+4x+13)^2)dx#?

# - 1/27pi #

Let:

Consider the first integral (we will deal with the improper integral)

Now, consider the second integral (again we will deal with the improper integral)

We can complete the square on the denominator to get:

I'll just quote this result here, as the integration is a bit tedious:

Combining our results we get:

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To solve the integral ( \int_{-\infty}^{+\infty} \frac{x}{(x^2 + 4x + 13)^2} , dx ), we can use the method of completing the square and partial fractions. After completing the square in the denominator, we can express the integrand as a sum of partial fractions. This allows us to integrate each term separately. The solution involves decomposition of the fraction into partial fractions and then integrating each term. The result is a combination of logarithmic and inverse trigonometric functions.

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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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