# Evaluate the definite integral x/√(e^x+(2+x)^2)

• MHB
• lfdahl
In summary, a definite integral is a mathematical concept used to calculate the area under a curve between two points on a graph. To evaluate a definite integral, one must identify the limits of integration and use techniques such as integration by parts or substitution to find the antiderivative of the function. The formula for evaluating the definite integral of x/√(e^x+(2+x)^2) is ∫[a, b] x/√(e^x+(2+x)^2) dx = (x*√(e^x+(2+x)^2) - 2ln|√(e^x+(2+x)^2) + x + 2|)[a, b]. It can
lfdahl
Gold Member
MHB
Evaluate

$$I = \int_{-2}^{0} \frac{x}{\sqrt{e^x+(2+x)^2}}\,dx$$

Suggested solution:

Use the substitution:

$v = (x+2)e^{-x/2}.$

$dv = -\frac{x}{2}e^{-x/2}dx$

$\frac{xdx}{\sqrt{e^x+(x+2)^2}} = \frac{xe^{-x/2}dx}{\sqrt{1+(x+2)^2e^{-x}}} = \frac{-2dv}{\sqrt{1+v^2}}$

Hence, the integral becomes:

$I = \int_{-2}^{0}\frac{xdx}{\sqrt{e^x+(x+2)^2}} = -2\int_{0}^{2}\frac{dv}{\sqrt{1+v^2}} = -2\sinh^{-1}(2)=e^{-2}-e^2.$

## 1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two points on a graph. It is used to calculate the total value of a function within a specific interval.

## 2. How do you evaluate a definite integral?

To evaluate a definite integral, you first need to identify the limits of integration, which are the points between which the area needs to be calculated. Then, you can use various techniques such as integration by parts or substitution to find the antiderivative of the function. Finally, you can plug in the limits of integration into the antiderivative to find the numerical value of the definite integral.

## 3. What is the formula for evaluating the definite integral of x/√(e^x+(2+x)^2)?

The formula for evaluating the definite integral of x/√(e^x+(2+x)^2) is ∫[a, b] x/√(e^x+(2+x)^2) dx = (x*√(e^x+(2+x)^2) - 2ln|√(e^x+(2+x)^2) + x + 2|)[a, b], where a and b are the limits of integration.

## 4. Can the definite integral of x/√(e^x+(2+x)^2) be solved analytically?

Yes, the definite integral of x/√(e^x+(2+x)^2) can be solved analytically by using various techniques such as substitution, integration by parts, or partial fractions. However, it may result in a complex solution and may require multiple steps to simplify the final answer.

## 5. What is the significance of evaluating the definite integral of x/√(e^x+(2+x)^2)?

Evaluating the definite integral of x/√(e^x+(2+x)^2) is significant as it helps in finding the total value of a function within a specific interval. This can be useful in various applications such as calculating work done, finding the average value of a function, or determining the area under a curve in real-life scenarios.

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