# How do you integrate #int 1/(xsqrtx)dx#?

The answer is

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Use power rule to integrate as shown below:

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To integrate ( \frac{1}{x \sqrt{x}} ), you can use substitution. Let ( u = \sqrt{x} ), then ( x = u^2 ) and ( du = \frac{1}{2\sqrt{x}} dx ). Substitute these into the integral:

[ \int \frac{1}{x \sqrt{x}} dx = \int \frac{1}{u^2} \cdot 2u , du = 2 \int u^{-1} , du ]

Now integrate ( u^{-1} ) with respect to ( u ):

[ 2 \int u^{-1} , du = 2 \ln|u| + C ]

Substitute back ( u = \sqrt{x} ):

[ 2 \ln|\sqrt{x}| + C = 2 \ln|x^{1/2}| + C = \boxed{\ln|x| + C} ]

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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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