# Integrating e^{\sqrt{x}}/\sqrt{x} from 1 to 4

• kartoshka
In summary, the conversation discusses solving the integral \int \frac{e^{\sqrt{x}}}{\sqrt{x}} using the substitution rule or symmetric function technique. The individual has tried different u substitutions, but none seem to work. They also ask about determining if a function is symmetric and whether it is even or odd. They mention that the integral is a definite integral from 1 to 4 and the top of the fraction is e^{\sqrt{x}}. They try the substitution u = \sqrt{x} again and end up with \int e^u * u^{-2}. They ask if there is a way to solve this without integration by parts and mention that they are typing on their phone without proper formatting. The
kartoshka
$\int$ $\frac{e^{\sqrt{x}}}{\sqrt{x}}$

It's in the substitution rule/symmetric function section of my book, so I figure I probably have to use one of those techniques to solve it. I've tried doing a bunch of different u substitutions $\sqrt{x}$, $e^{{\sqrt{x}}}$, etc, but none of them seem right.

How can you tell if a function is symmetric by looking at the equation? And whether it is even or odd?

PS - couldn't figure out how to do it, but it's actually a definite integral that goes from 1 to 4. Also, if the top of the fraction is hard to read, it's $e^{{\sqrt{x}}}$.

Try $$u = \sqrt{x}$$ again.

Well now I feel a bit ridiculous. But I end up with $/int$e^u * u^-2. Is there a way to solve this without integration by parts? We haven't gotten to it yet so I feel like there should be a way.

Sorry for the lack of formatting, typing on my phone and I can't remember most of the tags.

If $u = \sqrt{x}\,,$ then what is du ?

BTW: It's important to have the dx in the integral: $\displaystyle \int\frac{e^{\sqrt{x}}}{\sqrt{x}} dx\,.$

## What is the significance of integrating e^{\sqrt{x}}/\sqrt{x} from 1 to 4?

The integral of e^{\sqrt{x}}/\sqrt{x} from 1 to 4 is significant because it represents the area under the curve of the function e^{\sqrt{x}}/\sqrt{x} between the limits of 1 and 4. This area calculation has practical applications in various fields of science and technology.

## What is the mathematical expression for e^{\sqrt{x}}/\sqrt{x}?

The mathematical expression for e^{\sqrt{x}}/\sqrt{x} is a special type of exponential function known as the "exponential square root function". It can be written as e^{\sqrt{x}}/\sqrt{x} = (e^{\sqrt{x}})/(√x).

## What is the process for integrating e^{\sqrt{x}}/\sqrt{x}?

The process for integrating e^{\sqrt{x}}/\sqrt{x} involves using techniques such as substitution or integration by parts. The specific method used may vary depending on the complexity of the function. In this case, it is recommended to use substitution by letting u = √x and du = (1/2√x) dx.

## What is the result of integrating e^{\sqrt{x}}/\sqrt{x} from 1 to 4?

The result of integrating e^{\sqrt{x}}/\sqrt{x} from 1 to 4 is approximately 10.338. This can be calculated by substituting the limits into the antiderivative of e^{\sqrt{x}}/\sqrt{x} and evaluating the resulting expression.

## What are the practical applications of integrating e^{\sqrt{x}}/\sqrt{x}?

The integral of e^{\sqrt{x}}/\sqrt{x} has practical applications in fields such as physics, engineering, and economics. It can be used to calculate areas under curves in situations involving growth rates, population dynamics, and probability distributions. It also has applications in solving differential equations and modeling real-world scenarios.

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