# What is #f(x) = int -2x^2+1/xdx# if #f(2)=-1 #?

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To find the function ( f(x) = \int_{-2}^{x} -2x^2 + \frac{1}{x} dx ) given that ( f(2) = -1 ), we need to first integrate the given function and then use the provided information to solve for the constant of integration.

First, integrate the function:

( \int -2x^2 + \frac{1}{x} dx = -\frac{2}{3}x^3 + \ln|x| + C )

Next, substitute the limits of integration (-2 and x) into the integrated function:

( f(x) = -\frac{2}{3}x^3 + \ln|x| + C )

Using the given condition ( f(2) = -1 ):

( -1 = -\frac{2}{3}(2)^3 + \ln|2| + C )

Solve for ( C ):

( C = -\frac{16}{3} - \ln(2) )

Therefore, the function ( f(x) ) is:

( f(x) = -\frac{2}{3}x^3 + \ln|x| -\frac{16}{3} - \ln(2) )

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