# What is #f(x) = int e^(4x-1)-e^(3x-2)+e^x dx# if #f(2) = 3 #?

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To find ( f(x) = \int e^{4x-1} - e^{3x-2} + e^x , dx ) given that ( f(2) = 3 ), integrate each term of the function with respect to ( x ) and evaluate the integral at the upper limit of integration (which is ( x )) and subtract the value of the integral at the lower limit of integration (which is 0). Then, use the given condition ( f(2) = 3 ) to solve for the constant of integration. Finally, substitute the value of the constant of integration back into the function to obtain the final expression for ( f(x) ).

First, integrate each term: [ \int e^{4x-1} , dx = \frac{1}{4} e^{4x-1} ] [ \int e^{3x-2} , dx = \frac{1}{3} e^{3x-2} ] [ \int e^x , dx = e^x ]

Then, evaluate each integral at the upper limit of integration (which is ( x )) and subtract the value of the integral at the lower limit of integration (which is 0): [ \frac{1}{4} e^{4x-1} - \frac{1}{3} e^{3x-2} + e^x - (0) = \frac{1}{4} e^{4x-1} - \frac{1}{3} e^{3x-2} + e^x ]

Given that ( f(2) = 3 ), substitute ( x = 2 ) into the expression for ( f(x) ) and solve for the constant of integration: [ \frac{1}{4} e^{4(2)-1} - \frac{1}{3} e^{3(2)-2} + e^2 = 3 ]

[ \frac{1}{4} e^{7} - \frac{1}{3} e^{4} + e^2 = 3 ]

Finally, solve for the constant of integration: [ \frac{1}{4} e^{7} - \frac{1}{3} e^{4} + e^2 - 3 = 0 ]

[ \frac{1}{4} e^{7} - \frac{1}{3} e^{4} + e^2 = 3 ]

Therefore, the final expression for ( f(x) ) is: [ f(x) = \frac{1}{4} e^{4x-1} - \frac{1}{3} e^{3x-2} + e^x - 3 ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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