# How do you find the integral of #int dx/(x^2+2)^(3/2)# from negative infinity to infinity?

First, let's solve the indefinite integral with trig substitution:

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To find the integral of ( \int \frac{dx}{(x^2+2)^{3/2}} ) from negative infinity to infinity, you can use the method of residues from complex analysis. The integral can be represented as the contour integral of a complex function along a semicircular contour in the upper half-plane, which closes either above or below, depending on the sign of the exponent. Using the residue theorem, you can then find the residues of the function within the contour and sum them up. The integral over the semicircular arc tends to zero as the radius of the semicircle approaches infinity, leaving only the contribution from the residues. The result of the integral is then determined by the residues of the function at its singularities.

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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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