# What is #f(x) = int xsqrt(x-2) dx# if #f(2) = 3 #?

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To find ( f(x) = \int x\sqrt{x - 2} , dx ) when ( f(2) = 3 ), we need to solve the integral using integration by parts.

Let ( u = x ) and ( dv = \sqrt{x - 2} , dx ).

Then, ( du = dx ) and ( v = \frac{2}{3}(x - 2)^\frac{3}{2} ).

Using the integration by parts formula: [ \int u , dv = uv - \int v , du ]

[ \int x\sqrt{x - 2} , dx = \frac{2}{3}x(x - 2)^\frac{3}{2} - \int \frac{2}{3}(x - 2)^\frac{3}{2} , dx ]

[ = \frac{2}{3}x(x - 2)^\frac{3}{2} - \frac{2}{5}(x - 2)^\frac{5}{2} + C ]

Given that ( f(2) = 3 ), we can substitute ( x = 2 ) into the equation and solve for ( C ): [ 3 = \frac{2}{3}(2)(2 - 2)^\frac{3}{2} - \frac{2}{5}(2 - 2)^\frac{5}{2} + C ] [ 3 = 0 - 0 + C ] [ C = 3 ]

So, ( f(x) = \frac{2}{3}x(x - 2)^\frac{3}{2} - \frac{2}{5}(x - 2)^\frac{5}{2} + 3 ).

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