Chapter 5: Q27MP (page 275)

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

Short Answer

Expert verified

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

Step by step solution

Given Condition

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

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 $u : 0 \to 1$ and $v : 0 \to 1 + u$

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,

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

Evaluate the integral

Thus, the integral will be evaluated as follows,

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

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

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

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

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

Short Answer

Step by step solution

Given Condition

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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

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