# How do you evaluate the integral #int dx/sqrt(3-x)# from 1 to 3 if it converges?

We have:

We can use fractional and negative exponents in the integrand, as well as switching the direction of the bounds since we have the negative sign out front.

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we have a pattern here from the power and chain rules

and so we are integrating

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To evaluate the integral ( \int_{1}^{3} \frac{dx}{\sqrt{3-x}} ), we first need to check if it converges by examining the integrand at the endpoints of the interval [1, 3]. At ( x = 3 ), the denominator becomes 0, which indicates a potential issue with convergence. However, since the integrand is continuous and does not have a singularity in the interval [1, 3], the integral converges.

To evaluate the integral, we can use a substitution method. Let ( u = 3 - x ), then ( du = -dx ). The limits of integration change as follows: when ( x = 1 ), ( u = 3 - 1 = 2 ), and when ( x = 3 ), ( u = 3 - 3 = 0 ).

The integral becomes ( -\int_{2}^{0} \frac{du}{\sqrt{u}} ), which simplifies to ( -\int_{0}^{2} u^{-1/2} du ). Integrating ( u^{-1/2} ) with respect to ( u ) gives ( -2u^{1/2} ). Evaluating this from 0 to 2, we get ( -2(2^{1/2} - 0) ), which simplifies to ( -2\sqrt{2} ).

Therefore, the value of the integral is ( -2\sqrt{2} ).

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To evaluate the integral ( \int_{1}^{3} \frac{dx}{\sqrt{3-x}} ), we can use the substitution method. Let ( u = 3 - x ), then ( du = -dx ). Rewrite the integral in terms of ( u ), then integrate with respect to ( u ). Finally, revert back to the variable ( x ) to find the result.

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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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