# What is #f(x) = int (x+1)/((x+5)(x-4) ) dx# if #f(2)=1 #?

Decompose the rational function in partial fractions:

Then:

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To find ( f(x) ), integrate the given function ( \frac{x+1}{(x+5)(x-4)} ) with respect to ( x ). Then use the given condition ( f(2) = 1 ) to solve for the constant of integration.

First, decompose the rational function into partial fractions:

[ \frac{x+1}{(x+5)(x-4)} = \frac{A}{x+5} + \frac{B}{x-4} ]

Solve for ( A ) and ( B ):

[ x+1 = A(x-4) + B(x+5) ] [ x+1 = (A+B)x + (-4A+5B) ]

Equating coefficients:

[ A+B = 1 ] [ -4A+5B = 1 ]

Solve this system of equations to find ( A ) and ( B ):

[ A = \frac{-3}{9} ] [ B = \frac{4}{9} ]

So, the integral becomes:

[ \int \left( \frac{-3}{x+5} + \frac{4}{x-4} \right) dx ]

[ = -3 \ln|x+5| + 4 \ln|x-4| + C ]

Given that ( f(2) = 1 ), substitute ( x = 2 ) and solve for ( C ):

[ -3 \ln|2+5| + 4 \ln|2-4| + C = 1 ]

[ -3 \ln(7) + 4 \ln(-2) + C = 1 ]

[ C = 1 + 3 \ln(7) - 4 \ln(2) ]

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

[ f(x) = -3 \ln|x+5| + 4 \ln|x-4| + 1 + 3 \ln(7) - 4 \ln(2) ]

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