Q27MP Make the change of variables 聽t... [FREE SOLUTION]

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Chapter 5: Q27MP (page 275)

Make the change of variables to evaluate the integral ${鈭�}_{0}^{1} d x {鈭�}_{0}^{x} \frac{(x + y) e^{x + y}}{x^{2}} d y$

Short Answer

Expert verified

The integral is evaluated as ${鈭�}_{0}^{1} dx {鈭�}_{0}^{x} \frac{(x + y) e^{x + y}}{x^{2}} dy = e^{2} - e - 1$ .

Step by step solution

Given Condition

The integral given is ${鈭�}_{0}^{1} dx {鈭�}_{0}^{x} dy \frac{(x + y) e^{x + y}}{x^{2}} dy$ .

Concept of integration

Integration is a way of uniting the part to find a whole. Integration is used to define and calculate the area of the region bounded by the graph of functions.

Draw the diagram

Step 4: Apply the change of variables

The new boundaries of integration will be $u : 0 鈫� 1$ and $v : 0 鈫� 1 + u$

First, find $y (u, v)$ and $x (u, v)$

The values of $y = \frac{v u}{1 + u}$ and

$x = \frac{v}{1 + u}$ are known.

Step 5: Calculate the Jacobian determinant

The Jacobian determinant is found to be the following,

$|J| = |\begin{matrix} \frac{鈭� x}{鈭� v} & \frac{鈭� x}{鈭� u} \\ \frac{鈭� y}{鈭� v} & \frac{鈭� y}{鈭� u} \end{matrix}| = |\begin{matrix} \frac{1}{1 + u} & - \frac{v}{{(1 + u)}^{2}} \\ \frac{u}{1 + u} & \frac{v}{{(1 + u)}^{2}} \end{matrix}| = \frac{v}{{(1 + u)}^{3}} + \frac{v u}{{(1 + u)}^{3}} = \frac{v}{{(1 + u)}^{2}}$

Evaluate the integral

Thus, the integral will be evaluated as follows,

$l = {鈭�}_{0}^{1} d x {鈭�}_{0}^{x} d y \frac{(x + y) e^{x + y}}{x^{2}} d y = {鈭�}_{0}^{1} d u {鈭�}_{0}^{1 + u} \frac{v e^{v}}{v^{2}} {(1 + u)}^{2} \frac{v}{{(1 + u)}^{2}} d v = {鈭�}_{0}^{1} d u {鈭�}_{0}^{1 + u} d v e^{v} = {鈭�}_{0}^{1} (e^{1 + u} - 1) d u = (e^{1 + u} - 1)_{0}^{1} = e^{2} - e - 1$

Hence, ${鈭�}_{0}^{1} d x {鈭�}_{0}^{x} d y \frac{(x + y) e^{x + y}}{x^{2}} d y = e^{2} - e - 1$

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91影视

Make the change of variables to evaluate the integral ${鈭�}_{0}^{1} d x {鈭�}_{0}^{x} \frac{(x + y) e^{x + y}}{x^{2}} d y$

Short Answer

Step by step solution

Given Condition

Concept of integration

Draw the diagram

Step 4: Apply the change of variables

Step 5: Calculate the Jacobian determinant

Evaluate the integral

One App. One Place for Learning.

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Make the change of variables to evaluate the integral鈭�01dx鈭�0x(x+y)ex+yx2dy

Short Answer

Step by step solution

Given Condition

Concept of integration

Draw the diagram

Step 4: Apply the change of variables

Step 5: Calculate the Jacobian determinant

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Measurements

Further Mechanics and Thermal Physics

Famous Physicists

Solid State Physics

Electromagnetism

Study anywhere. Anytime. Across all devices.

Make the change of variables to evaluate the integral ${鈭�}_{0}^{1} d x {鈭�}_{0}^{x} \frac{(x + y) e^{x + y}}{x^{2}} d y$