# How do you integrate #ln(x) x^ (3/2) dx#?

For this problem, we will use integration by parts:

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To integrate (\ln(x) \cdot x^{3/2} , dx), you can use integration by parts. Let (u = \ln(x)) and (dv = x^{3/2} , dx). Then, (du = \frac{1}{x} , dx) and (v = \frac{2}{5} x^{5/2}).

Applying integration by parts formula:

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

yields:

[ \int \ln(x) \cdot x^{3/2} , dx = \frac{2}{5} x^{5/2} \ln(x) - \int \frac{2}{5} x^{5/2} \cdot \frac{1}{x} , dx ]

Simplify and integrate:

[ \int \ln(x) \cdot x^{3/2} , dx = \frac{2}{5} x^{5/2} \ln(x) - \frac{2}{5} \int x^{3/2} , dx ]

[ = \frac{2}{5} x^{5/2} \ln(x) - \frac{2}{5} \cdot \frac{2}{5} x^{5/2} + C ]

[ = \frac{2}{5} x^{5/2} \left( \ln(x) - \frac{2}{5} \right) + C ]

where (C) is the constant of integration.

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