find the antiderivative that satisfies the given condition `dy/dx= (8*x^2*e^(2x) - 5x)/x^2` and `y(1)=0`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function has to be determined for which `dy/dx= (8x^2*(e^(2x) - 5x))/x^2`and y(1)=0

`dy/dx= (8x^2*e^(2x) - 5x)/x^2`

=> `dy = (8*e^(2x) - 5/x) dx`

`y = int (8*e^(2x) - 5/x) dx`

=> `8*e^(2x)/2 -5*ln x + C`

=> `4*e^(2x) - 5*ln x + C`

y(1) = 0

=> `4*e^(2x) - 5*ln 1 + C = 0`

=> `C = -4*e^2`

This gives `y = 4*e^(2x) - 5*ln x - 4*e^2`

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