# How do you integrate # ln x / x^(1/2)#?

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To integrate ( \frac{\ln x}{\sqrt{x}} ), you can use integration by parts. Let ( u = \ln x ) and ( dv = x^{-1/2} , dx ).

[ du = \frac{1}{x} , dx ] [ v = \frac{x^{1/2}}{1/2} = 2x^{1/2} ]

Now, apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substituting the values:

[ \int \frac{\ln x}{\sqrt{x}} , dx = 2x^{1/2} \ln x - \int 2x^{1/2} \frac{1}{x} , dx ]

[ = 2x^{1/2} \ln x - 2 \int x^{1/2 - 1} , dx ]

[ = 2x^{1/2} \ln x - 2 \int x^{-1/2} , dx ]

[ = 2x^{1/2} \ln x - 2(2x^{1/2}) + C ]

[ = 2x^{1/2} (\ln x - 1) + 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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