- #1
steffen ecca
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Homework Statement
Waht is the correct integral of
Homework Equations
2^{-x}
The Attempt at a Solution
Is -{1/(x-1)}*2^{1-x} correct or do I have to apply some substitution?
Thanks for your answers!
You need to learn very quickly and very thoroughly that the derivative formula [itex]d(x^n)/dx= nx^{n-1}[/itex] and the corresponding integral formula [itex]\int x^n dx= 1/(n+1) x^{n+1}+ C[/itex] only hold when the variable, x, is the base and the exponent is a constant. The situation in which the base is a constant and the exponent is x, is completely different. Both derivative and integral, for example, of [itex]e^x[/itex] is just [itex]e^x[/itex] itself (plus "C" for the integral, of course).steffen ecca said:Homework Statement
Waht is the correct integral of
Homework Equations
2^{-x}
The Attempt at a Solution
Is -{1/(x-1)}*2^{1-x} correct or do I have to apply some substitution?
Thanks for your answers!
steffen ecca said:Homework Statement
Waht is the correct integral of
Homework Equations
2^{-x}
The Attempt at a Solution
Is -{1/(x-1)}*2^{1-x} correct or do I have to apply some substitution?
Thanks for your answers!
The formula for integration of 2^{-x} is 1/(ln2)^{2} e^{-x} + C, where C is the constant of integration.
The process for integrating 2^{-x} involves using the power rule for integration, followed by substituting the value of x into the formula and solving for the constant of integration.
The domain of 2^{-x} is all real numbers, while the range is (0, infinity).
Integration of 2^{-x} is commonly used in finance and economics to model exponential decay and growth, as well as in physics to calculate the rate of change in radioactive decay.
As x increases, the graph of 2^{-x} approaches the x-axis but never touches it. It also decreases rapidly, approaching 0 as x approaches infinity.