ExpIntegrals

ExpIntegrals - previous index next Some Useful Integrals of...

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previous index next Some Useful Integrals of Exponential Functions Michael Fowler We’ve shown that differentiating the exponential function just multiplies it by the constant in the exponent, that is to say, . ax ax d ea e dx = Integrating the exponential function, of course, has the opposite effect: it divides by the constant in the exponent: 1 , ax ax ed x e a = as you can easily check by differentiating both sides of the equation. An important definite integral (one with limits) is 0 1 . ax x a = Notice the minus sign in the exponent: we need an integrand that decreases as x goes towards infinity, otherwise the integral will itself be infinite. To visualize this result, we plot below e - x and e - 3 x . Note that the green line forms the hypotenuse of a right-angled triangle of area 1, and it is very plausible from the graph that the total area under the e x curve is the same, that is, 1, as it must be. The e - 3 x curve has area 1/3 under it, ( a = 3).
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2 Now for something a bit more challenging: how do we evaluate the integral () 2 ? ax I ae d x −∞ = ( a has to be positive, of course.) The integral will definitely not be infinite: it falls off equally fast in both positive and negative directions, and in the positive direction for x greater than 1, it’s smaller than e - ax , which we know converges. To see better what this function looks like, we plot it below for a = 1 (red) and a = 4 (blue).
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3 Notice first how much faster than the ordinary exponential e - x this function falls away. Then note that the blue curve, a
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ExpIntegrals - previous index next Some Useful Integrals of...

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