Chapter 5: Q5P (page 247)

Evaluate the integrals ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x$ .

Short Answer

Expert verified

The value of given integral is ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x i s \frac{3 e^{2} - 5}{12}$ .

Step by step solution

Given data

The given equation is ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x$ .

Concept of Partial differential equation

Integration is a technique of finding a function $g (x)$ the derivative of which, $D g (x)$ , is equal to a given function $f (x)$ .

This is indicated by the integral sign " $鈭�$ , as in $鈭� f (x)$ , usually called the indefinite integral of the function.

Differentiate the equation

Consider the equation as follows:

${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = {鈭�}_{x = 0}^{1} d x ({鈭�}_{y = x}^{e^{x}} y d y) {鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = {鈭�}_{x = 0}^{1} d x {(\frac{y^{2}}{2})}_{y = x}^{e^{x}} {鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = \frac{1}{2} {鈭�}_{x = 0}^{1} (e^{2 x} - x^{2}) d x {鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = \frac{1}{2} {(\frac{e^{2 x}}{2} - \frac{x^{3}}{3})}_{0}^{1}$

Further, solve the equation as follows:

${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = \frac{1}{4} (e^{2} - 1) - \frac{1}{6} {鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = \frac{3 e^{2} - 5}{12}$

Therefore, the value of given integral is ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x = \frac{3 e^{2} - 5}{12}$ .

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91影视

Evaluate the integrals ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x$ .

Short Answer

Step by step solution

Given data

Concept of Partial differential equation

Differentiate the equation

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Short Answer

Step by step solution

Given data

Concept of Partial differential equation

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Evaluate the integrals ${鈭�}_{x = 0}^{1} {鈭�}_{y = x}^{e^{x}} y d y d x$ .