# What is #f(x) = int x^3-2x dx# if #f(-2)=-1 #?

Now applying the initial boundary condition we get

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To find ( f(x) = \int (x^3 - 2x) , dx ) if ( f(-2) = -1 ), integrate the function and then use the given condition to find the constant of integration ( C ).

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

Given that ( f(-2) = -1 ):

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

[ \frac{16}{4} - 4 + C = -1 ]

[ 4 - 4 + C = -1 ]

[ C = -1 + 4 - 4 = -1 ]

So, ( C = -1 ).

Therefore, ( f(x) = \frac{x^4}{4} - x^2 - 1 ).

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