# What is #f(x) = int (x-2)(e^x-1) dx# if #f(2 ) = 4 #?

Develop the integrand:

The only term that not immediate is the first, that can be integrated by parts:

So:

Finally:

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To find ( f(x) ), integrate the function ( (x-2)(e^x-1) ) with respect to ( x ) using the given condition ( f(2) = 4 ).

[ f(x) = \int (x-2)(e^x-1) , dx ]

Given that ( f(2) = 4 ), we have:

[ 4 = \int_{2}^{2} (x-2)(e^x-1) , dx ]

[= 0 ]

Thus, ( f(x) = 0 ).

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