What is #f(x) = int e^(2x1)e^(3x2)+e^x dx# if #f(2) = 3 #?
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To find ( f(x) = \int e^{2x1}  e^{3x2} + e^x , dx ) given ( f(2) = 3 ), you need to find the indefinite integral of the function and then use the given condition to solve for the constant of integration. However, the given function is a sum of exponentials, so you can integrate each term separately and then add them up. After integrating each term, you can plug in the upper limit ( x = 2 ) and solve for the constant of integration to find the value of ( f(2) ).
Let's integrate each term separately:

Integral of ( e^{2x1} ): [ \int e^{2x1} , dx = \frac{1}{2}e^{2x1} + C_1 ]

Integral of ( e^{3x2} ): [ \int e^{3x2} , dx = \frac{1}{3}e^{3x2} + C_2 ]

Integral of ( e^x ): [ \int e^x , dx = e^x + C_3 ]
Now, summing up the integrals with their respective constants of integration: [ f(x) = \frac{1}{2}e^{2x1}  \frac{1}{3}e^{3x2} + e^x + C ]
Now, use ( f(2) = 3 ) to solve for ( C ): [ f(2) = \frac{1}{2}e^{2(2)1}  \frac{1}{3}e^{3(2)2} + e^2 + C = 3 ] [ \frac{1}{2}e^3  \frac{1}{3}e^4 + e^2 + C = 3 ] [ \frac{1}{2}e^3  \frac{1}{3}e^4 + e^2 + C  3 = 0 ]
Solve for ( C ): [ C = 3  \frac{1}{2}e^3 + \frac{1}{3}e^4  e^2 ]
So, the value of the constant of integration is: [ C = 3  \frac{1}{2}e^3 + \frac{1}{3}e^4  e^2 ]
Thus, the function ( f(x) ) is: [ f(x) = \frac{1}{2}e^{2x1}  \frac{1}{3}e^{3x2} + e^x + 3  \frac{1}{2}e^3 + \frac{1}{3}e^4  e^2 ]
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