# What is #f(x) = int (x-1)^3 dx# if #f(-1) = 1 #?

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To find the function ( f(x) = \int (x - 1)^3 , dx ) given ( f(-1) = 1 ), we need to integrate ( (x - 1)^3 ) and then find the constant of integration using the given information.

The indefinite integral of ( (x - 1)^3 ) is ( \frac{1}{4}(x - 1)^4 + C ), where ( C ) is the constant of integration.

Now, to find the value of ( C ), we use the given condition ( f(-1) = 1 ).

So, ( f(-1) = \frac{1}{4}(-1 - 1)^4 + C = 1 ).

Solving this equation for ( C ), we get:

[ \frac{1}{4}(0)^4 + C = 1 ] [ C = 1 ]

Thus, the function ( f(x) ) is given by:

[ f(x) = \frac{1}{4}(x - 1)^4 + 1 ]

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