How do you show whether the improper integral #int dx / (x^2 + sinx)# converges or diverges from 2 to infinity?

Answer 1

#int_2^oo dx / (x^2 + sinx) lelog_e(sqrt(3))#

#int dx / (x^2 + sinx)le int dx/(x^2-1)#

so

#int_2^oo dx / (x^2 + sinx) lelim_{x->oo}log_e sqrt(abs(1-x)/abs(1+x)) +log_e(sqrt(3))#

so

#int_2^oo dx / (x^2 + sinx) lelog_e(sqrt(3))#
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Answer 2

To determine the convergence or divergence of the improper integral ∫(2 to ∞) dx / (x^2 + sin(x)), you can use the limit comparison test. First, note that the integral is improper because it extends to infinity.

  1. Find a function that behaves similarly to the given function as x approaches infinity. In this case, choose f(x) = 1/x^2 because it's simpler to integrate and has similar behavior to 1/(x^2 + sin(x)) as x approaches infinity.

  2. Compute the limit as x approaches infinity of the quotient of the two functions, lim (x→∞) [(1/x^2) / (1/(x^2 + sin(x)))].

  3. Simplify the expression and determine if the limit is finite and nonzero. If the limit is finite and nonzero, then the given integral and the chosen function have the same convergence behavior. If the limit is zero or infinite, choose another comparison function.

  4. Integrate the chosen function over the given interval and determine its convergence behavior. In this case, ∫(2 to ∞) dx / (x^2) is a standard integral that converges.

  5. Since the chosen function converges, and it behaves similarly to the given function as x approaches infinity, by the limit comparison test, the given integral also converges.

Therefore, the improper integral ∫(2 to ∞) dx / (x^2 + sin(x)) converges.

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Answer from HIX Tutor

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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