# What is #f(x) = int 3x^3+xe^(x-2) dx# if #f(2 ) = 3 #?

by Parts (IBP).

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To find ( f(x) = \int (3x^3 + xe^{x-2}) , dx ) if ( f(2) = 3 ), we need to first integrate the function and then evaluate it using the given condition.

Given ( f(2) = 3 ), we integrate ( 3x^3 + xe^{x-2} ) with respect to ( x ):

[ \int (3x^3 + xe^{x-2}) , dx = \frac{3}{4}x^4 + \int xe^{x-2} , dx ]

To integrate ( \int xe^{x-2} , dx ), we can use integration by parts:

[ u = x, \quad dv = e^{x-2} , dx ]

[ du = dx, \quad v = e^{x-2} ]

Using the integration by parts formula:

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

[ \int xe^{x-2} , dx = xe^{x-2} - \int e^{x-2} , dx ]

[ = xe^{x-2} - e^{x-2} + C ]

Now, substitute this result back into the integral:

[ \frac{3}{4}x^4 + xe^{x-2} - e^{x-2} + C ]

Now, we'll use the condition ( f(2) = 3 ):

[ 3 = \frac{3}{4}(2)^4 + 2e^{2-2} - e^{2-2} + C ]

[ 3 = \frac{3}{4}(16) + 2 - 1 + C ]

[ 3 = 12 + 2 - 1 + C ]

[ 3 = 13 + C ]

[ C = -10 ]

Therefore, ( f(x) = \frac{3}{4}x^4 + xe^{x-2} - e^{x-2} - 10 ).

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