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Cerrena [4.2K]
3 years ago
8

Primitive function for: 1. f(x)=4e2x+e−x, sådan att F(0)=2

Mathematics
1 answer:
Free_Kalibri [48]3 years ago
6 0

The primitive function for f(x) = 4e^{2x} + e^{-x} is given by it's indefinite integral.

To calculate it, recall:

\frac{d}{dx}e^{ax}=ae^{ax}

\int c f(x)dx = c\int f(x)dx

Let's begin

\int 4e^{2x}+e^{-x}dx \\4\int e^{2x}dx+ \int e^{-x}dx\\4\frac{e^{2x}}{2}+\frac{e^{-x}}{-1}  + c\\

F(x) = 2e^{2x}-e^{-x} +c

To find the costant, we just need to use the fact that F(0)=2

F(0) = 2 \\4e^{0}+e^{0} + c =2\\5+c =2\\c=-3

Therefore,

\boxed{F(x) = 2e^{2x} - e^{-x} -3 }

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