How to find standard deviation of a sample?

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Ответ:

MannyBanko1350

18.03.2020

Mathematics

The improper integral converges and $\int_0^{-\infty} e^{7x}dx = -\frac{1}{7}$ .

Step-by-step explanation:

First, I assume that the integral in question is

$\int_0^{-\infty} e^{7x}dx$ .

Now, the integral is improper because, at least, one of the limits is $\pm\infty$ . We need to recall that an improper integral

$\int_0^{-\infty} f(x)dx$

converges, by definition, if the following limit exist:

$\lim_{A\rightarrow -\infty} \int_0^A f(x)dx = \int_0^{-\infty} f(x)dx$ .

In this particular case we need to study the limit

$\lim_{A\rightarrow -\infty} \int_0^A e^{7x}dx$ .

In order to complete this task we calculate the integral $\int_0^A e^{7x}dx$ . Then,

$\int_0^A e^{7x}dx = \frac{e^{7x}}{7}\Big|_0^A = \frac{e^{7A}}{7} - \frac{1}{7}$ .

Substituting the above expression into the limit we have

$\lim_{A\rightarrow -\infty} \frac{e^{7A}}{7} - \frac{1}{7} = - \frac{1}{7}$

because

$\lim_{A\rightarrow -\infty} \frac{e^{7A}}{7}=0$ .

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