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I am not sure how to evaluate this problem

$$\int_{3}^{\infty} \frac{dx}{(x-2)^\frac{3}{2}}$$

I do not know if it converges or diverges but I do know that it is undefined at $3$.

It is $3$ not $-3$.

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2 Answers 2

up vote 5 down vote accepted

The integrand is defined for any $x\ge3$. The integral is improper, however, since the upper limit of integration is infinite. You'll have to set it up as a limit: $$ \int_{3}^\infty{dx\over (x-2)^{3/2}} =\lim_{a\rightarrow\infty}\int_{3}^a{dx\over (x-2)^{3/2}}. $$ Evaluate $\int_{3}^a{dx\over (x-2)^{3/2}}$ first. This will leave you with an expression in $a$. Then you can take the limit. If the limit exists, the integral converges to its value. If the limit does not exist (including in the infinite sense), then the integral diverges.

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$$\int_3^\infty {dx\over(x-2)^{3/2}} = \int_1^\infty {dx\over x^{3/2}} = \lim_{r\to\infty} 2 - {2\over \sqrt{r}} = 2. $$

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