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The answer is ln (V) + C

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Rainy Farkya

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10mo ago

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To convert a divergence to a surface integral, you can use the Divergence Theorem, which states that for a vector field (\mathbf{F}) defined in a region (V) with a smooth boundary surface (S), the integral of the divergence of (\mathbf{F}) over (V) is equal to the flux of (\mathbf{F}) across (S). Mathematically, this is expressed as: [ \int_V (\nabla \cdot \mathbf{F}) , dV = \iint_S \mathbf{F} \cdot \mathbf{n} , dS ] where (\mathbf{n}) is the outward unit normal to the surface (S). Thus, you can transform a volume integral of divergence into a surface integral by applying this theorem.


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Integral of xlnxdx?

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Integration of xtanx?

To integrate ( x \tan(x) ), we can use integration by parts. Let ( u = x ) and ( dv = \tan(x) , dx ). This gives ( du = dx ) and ( v = -\ln|\cos(x)| ). Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), we obtain: [ \int x \tan(x) , dx = -x \ln|\cos(x)| + \int \ln|\cos(x)| , dx + C ] The integral ( \int \ln|\cos(x)| , dx ) does not have a simple closed form, so the final result may be expressed in terms of this integral along with the logarithmic term.