Evaluate the integral integral 0 to infinity x n e -x dx , for n= 0,1,2,3

Answered: Evaluate the integral integral 0 to…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter7: Integration

Section7.3: Area And The Definite Integral

Problem 21E

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Question

Evaluate the integral

integral 0 to infinity

x ⁿe ^{-x dx} , for n= 0,1,2,3

Expert Solution

Step 1

We are given the function $f (x) = x^{n} e^{- x}$ .

Our aim is to find the integral of f(x) from 0 to infinity for n = 0, 1, 2, 3.

Step 2

Take n = 0.

Thus, integral can be evaluated as follows;

$\begin{array}{rcl} \int_{0}^{\infty} e^{- x} d x & = & - [e^{- x}] \begin{matrix} \infty \\ 0 \end{matrix} \\ = & - [e^{- \infty} - e^{0}] \\ = & - [0 - 1] \\ = & 1 \end{array}$

Taking, n = 1, the integral value will be,

$\begin{array}{rcl} \int_{0}^{\infty} x e^{- x} d x & = & [- x e^{- x}] \begin{matrix} \infty \\ 0 \end{matrix} + \int_{0}^{\infty} e^{- x} d x \\ = & 0 + 1 \\ = & 1 \end{array}$

Taking, n = 2, the integral value will be,

$\begin{array}{rcl} \int_{0}^{\infty} x^{2} e^{- x} d x & = & [- x^{2} e^{- x}] \begin{matrix} \infty \\ 0 \end{matrix} + 2 \int_{0}^{\infty} x e^{- x} d x \\ = & 0 + 2 (1) \\ = & 2 \end{array}$

Taking, n = 3, the integral value will be,

$\begin{array}{rcl} \int_{0}^{\infty} x^{3} e^{- x} d x & = & [- x^{3} e^{- x}] \begin{matrix} \infty \\ 0 \end{matrix} + 3 \int_{0}^{\infty} x^{2} e^{- x} d x \\ = & 0 + 3 (2) \\ = & 6 \end{array}$

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