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

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

ReportAnswer 1

$f \left(x\right) = \frac{1}{2} {e}^{2 x - 1} - \frac{1}{3} {e}^{3 x - 2} + {e}^{x} + 3.768$

#### Explanation:

First integrate to obtain $f \left(x\right) = \frac{1}{2} {e}^{2 x - 1} - \frac{1}{3} {e}^{3 x - 2} + {e}^{x} + c$
Then substitute $f \left(2\right) = 3$ to obtain $3 = \frac{1}{2} {e}^{3} - \frac{1}{3} {e}^{4} + {e}^{2} + c$
Rearrange for c then evaluate: $c = 3 - \frac{1}{2} {e}^{3} + \frac{1}{3} {e}^{4} - {e}^{2} = 3.768$ (4s.f.)
Thus the original function is $f \left(x\right) = \frac{1}{2} {e}^{2 x - 1} - \frac{1}{3} {e}^{3 x - 2} + {e}^{x} + 3.768$

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