# How do you find the antiderivative of #2/(xsqrtx)#?

Rewrite it as

Examine your response by making a distinction.

Try writing the solution without using any negative exponents.

In the absence of logical exponents:

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

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

Simplify:

[ \int \frac{4}{u^3} , du ]

Now integrate with respect to ( u ):

[ \int \frac{4}{u^3} , du = 4 \int u^{-3} , du = 4 \left( \frac{u^{-2}}{-2} \right) + C ]

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

[ = -\frac{2}{\sqrt{x}^2} + C = -\frac{2}{x} + C ]

So, the antiderivative of ( \frac{2}{x\sqrt{x}} ) is ( -\frac{2}{x} + C ), where ( C ) is the constant of integration.

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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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