calculation integral of 1 / e^(x/2) dx

Introduction

Integration is one of the fundamental concepts in mathematics used to calculate the area under the curve of a function. One of the common types of integrals is the definite integral, which has specific upper and lower bounds. In this article, we will discuss how to calculate the integral of the function $\frac{1}{{�}^{�\mathrm{/}2}}$ from $0$ to $\mathrm{\infty }$. This function appears in various scientific contexts, including physics, economics, and statistics.

Part 1: Introduction to Integration

Before understanding how to calculate a definite integral, let's discuss the basic concept of integration. Integration is a mathematical operation used to find the area under the curve of a function. It is useful in various situations, such as calculating the area under the graph of a function, determining the total accumulation of a quantity, or finding the average value of a function.

There are two main types of integrals: definite integrals and indefinite integrals. Definite integrals have given upper and lower bounds, as we encounter in this problem, while indefinite integrals seek the antiderivative of a function.

Part 2: Integral of $\frac{1}{{�}^{�\mathrm{/}2}}$

In this problem, we want to calculate the integral of the function $\frac{1}{{�}^{�\mathrm{/}2}}$ from $0$ to $\mathrm{\infty }$. Let's start by determining the antiderivative of this function.

The antiderivative of $\frac{1}{{�}^{�\mathrm{/}2}}$ can be found using substitution. Let's make the substitution $�=\frac{�}{2}$, so $��=2��$. Then, we can transform the original integral into the following form:

$$\int \frac{1}{{�}^{�\mathrm{/}2}}��=\int {�}^{-�}\cdot 2��=2\int {�}^{-�}��$$

Now, we can calculate the integral of ${�}^{-�}$:

$$2\int {�}^{-�}��=-2{�}^{-�}+�$$

where $�$ is the constant of integration.

Part 3: Computing the Definite Integral

Now that we have the antiderivative of the function $\frac{1}{{�}^{�\mathrm{/}2}}$, which is $-2{�}^{-�}+�$, we will proceed to calculate the definite integral from $0$ to $\mathrm{\infty }$:

$${\int }_0^{\mathrm{\infty }}\frac{1}{{�}^{�\mathrm{/}2}}��=\underset{�\to \mathrm{\infty }}{\lim }(-2{�}^{-�}){\mid }_0^{�}$$

As $�$ approaches $\mathrm{\infty }$, ${�}^{-�}$ approaches $0$, and when $�=0$, ${�}^{-�}=1$. So, the integral becomes:

$$\underset{�\to \mathrm{\infty }}{\lim }(-2{�}^{-�}-(-2{�}^0))=-2(0-(-2))=-2(2)=-4$$

Part 4: Conclusion

In this article, we have explained the concept of integration, specifically definite integration, which is used to calculate the area under the curve of a function. Furthermore, we calculated the integral of the function $\frac{1}{{�}^{�\mathrm{/}2}}$ from $0$ to $\mathrm{\infty }$, using substitution and finding its antiderivative. The result of the integral is 2.

Integration is a highly valuable tool in various fields of science and engineering, and understanding how to calculate it is an important mathematical skill. By mastering the concept of integration, we can tackle a wide range of mathematical problems that arise in everyday life and scientific research.