Proof Prove that where Then evaluate each integral. (a) (b) (c)

Chapter 8, Problem 104

(choose chapter or problem)

Proof Prove that \(I_{n}=\left(\frac{n-1}{n+2}\right) I_{n-1}\), where

\(I_{n}=\int_{0}^{\infty} \frac{x^{2 n-1}}{\left(x^{2}+1\right)^{n+3}} d x, \quad n \geq 1\).

Then evaluate each integral.

(a) \(\int_{0}^{\infty} \frac{x}{\left(x^{2}+1\right)^{4}} d x\)

(b) \(\int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{5}} d x\)

(c) \(\int_{0}^{\infty} \frac{x^{5}}{\left(x^{2}+1\right)^{6}} d x\)

Text Transcription:

I_n=(n-1/n+2)I_n-1

I_n=Int_0^infinity x^2n-1/(x^2+1)^n+3 dx, n geq 1

Int_0^infinity x/(x^2+1)^4 dx

Int_0^infinity x/(x^2+1)^5 dx

Int_0^infinity x/(x^2+1)^6 dx