   Chapter 8.2, Problem 66E

Chapter
Section
Textbook Problem

# Finding a General Rule In Exercises 69 and 70, use a computer algebra system to find theintegrals for n = 0, 1, 2,and 3. Use the result to obtain a general rule for the integrals for anypositive integer n and test your results for n = 4. ∫ x n e x d x

To determine

To calculate: the value of integral xnexdx using a Computer Algebra System and also obtain a general rule.

Explanation

Given: The integral xnexdx and for n = 0, 1, 2, 3, and 4.

Formula Used: the integration of exponential function is given as:

exdx=ex+C

Calculation:

For n = 0, the integral xnexdx can be written as to exdx.

exdx=ex+C

Also by using a Computer Algebra System, the value of integral be verified as,

———-(1)

Now, For n = 1, the integral xnexdx can be written as to xexdx

On Integrating by parts

Let u=x and dv=exdx, such that

On differentiating u with respect to x

We get,

du=dx——(2)

On integrating v we get,

v=exdx=ex——(3)

Now,

xexdx=xexexdx=xexex+C

Also by using a Computer Algebra System, the value of integral xexdx

Be verified as:

———(4)

Now, For n = 2, the integral xnexdx can be written as to x2exdx

On Integrating by parts

Let u=x2 and dv=exdx, such that

On differentiating u and integrating v

We get,

du=2xdx——(5)

v=exdx=ex——(6)

Now, Using (4)

x2exdx=x2ex2xexdx=x2ex2(xexex)+C=ex(x22x2)+C

Also by using a Computer Algebra System, the value of integral x2exdx

Be verified as;

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