# How do you test the improper integral #int x/absxdx# from #[-5,3]# and evaluate if possible?

To test the improper integral (\int_{-5}^{3} \frac{x}{|x|} , dx), we need to examine the behavior of the integrand as it approaches the endpoints of the interval ([-5, 3]).

Since the function (\frac{x}{|x|}) is not continuous at (x = 0), we split the integral into two parts: (\int_{-5}^{0} \frac{x}{|x|} , dx) and (\int_{0}^{3} \frac{x}{|x|} , dx).

For the first part, (\int_{-5}^{0} \frac{x}{|x|} , dx), we have: [\int_{-5}^{0} \frac{x}{|x|} , dx = \int_{-5}^{0} -1 , dx = -\int_{-5}^{0} 1 , dx = -(-5 - 0) = 5.]

For the second part, (\int_{0}^{3} \frac{x}{|x|} , dx), we have: [\int_{0}^{3} \frac{x}{|x|} , dx = \int_{0}^{3} 1 , dx = 3 - 0 = 3.]

Therefore, the integral (\int_{-5}^{3} \frac{x}{|x|} , dx) converges, and its value is (5 + 3 = 8).

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The integral does exist and equals

Now

and

Therefore, these integrals converge and

In the end, you should know that a finite number of "jump discontinuities " over a finite interval will not cause an improper integral over that interval to diverge.

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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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