   Chapter 8.8, Problem 100E

Chapter
Section
Textbook Problem

Proof Prove that I n = ( n − 1 n + 2 ) I n − 1 ' where I n = ∫ 0 ∞ x 2 n − 1 ( x 2 + 1 ) n + 3 d x , n ≥ 1. Then evaluate each integral. ( a )   ∫ 0 ∞ x ( x 2 + 1 ) 4 d x (b) ∫ 0 ∞ x 3 ( x 2 + 1 ) 5 d x (c)   ∫ 0 ∞ x 5 ( x 2 + 1 ) 6 d x

To determine

To Prove: In=(n1n+2)In1, where In=0x2n1(x2+1)n+3dx

Explanation

Given:

The integral for In is given as:

In=0x2n1(x2+1)n+3dx

Where, n1.

Formula used:

To check for convergence, the following limit must exist.

abf(x)dx=limbabf(x)dx

According to the by parts formula:

I=uvvu.

Proof:

As, n1, it is first checked for n=1.

I1=0x(x2+1)4dx=12limb0b2x(x2+1)4dx

If the function, u=g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, then

abf(g(x))g(x)dx=g(a)g(b)f(u)du

Here, u=(x2+1),du=2xdx

(a)

To determine

To calculate: The value of the integral 0x(x2+1)4dx.

(b)

To determine

To calculate: The value of the integral 0x3(x2+1)5dx

(c)

To determine

To calculate: The value of the integral 0x5(x2+1)6dx.

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