# Validity of Reduction Formula

by jimbobian
Tags: formula, reduction, validity
 P: 52 1. The problem statement, all variables and given/known data If In denotes $$\int_0^∞ \! \frac{1}{(1+x^2)^n} \, \mathrm{d} x$$ Prove that $2nI_{n+1} = (2n-1)I_n$, and state the values of n for which this reduction formula is valid. 2. Relevant equations 3. The attempt at a solution $$I_n=\int_0^∞ \! \frac{1}{(1+x^2)^n} \, \mathrm{d} x$$ $$=\int_0^∞ \! (1+x^2)^{-n} \, \mathrm{d} x$$ By parts: $$=\left[ x(1+x^2)^{-n} \right]_0^∞ + 2n\int_0^∞ \! \frac{x^2}{(1+x^2)^{n+1}} \, \mathrm{d} x$$ $$=0 + 2n\int_0^∞ \! \frac{(1+x^2)-1}{(1+x^2)^{n+1}} \, \mathrm{d} x$$ $$=2n\int_0^∞ \! \frac{(1+x^2)}{(1+x^2)^{n+1}} \, \mathrm{d} x - 2n\int_0^∞ \! \frac{1}{(1+x^2)^{n+1}} \, \mathrm{d} x$$ $$=2nI_{n}-2nI_{n+1}$$ $$2nI_{n+1}=(2n-1)I_{n}$$ as required. It's the next bit where I'm stuck - the range of values for which n is valid. Obviously when part of the integral has been evaluated (following parts), this requires that n>0 otherwise the expression doesn't converge. I can't see anywhere else in the method where there is a restriction for n to be a specific value so I went with n>0 as my answer, but my book says n>1/2, can anyway shed some light on this for me. Thanks
 P: 1,020 the expression inside the integral is always positive and so does the integral.at n=1/2 the integral does not converge and observe that at n=1/2 ,I(n+1) is zero and for 0
 P: 3,014 you should use: $$\frac{1}{(1 + x^2)^n} = \frac{1 + x^2 - x^2}{(1 + x^2)^n} = \frac{1}{(1 + x^2)^{n - 1}} - \frac{x^2}{(1 + x^2)^n}$$ For the integral of the second term, use integration by parts: $$-\int_{0}^{\infty}{x \, \frac{x}{(1 + x^2)^n} \, dx}$$ $$u = x \Rightarrow du = dx$$ $$dv = \frac{x}{(1 + x^2)^n} \, dx \Rightarrow v = \int{ \frac{x}{(1 + x^2)^n} \, dx} \stackrel{t = 1 + x^2}{=} \frac{1}{2} \, \int{t^{-n} \, dt} = \frac{t^{1- n}}{2(1 - n)} = -\frac{1}{2 (n - 1) (1 + x^2)^{n - 1}}$$ Combine everything, identify the relevant integrals with $I_n$, and $I_{n - 1}$, and see what you get.
 P: 3,014 Validity of Reduction Formula Oh, I see you already did the steps. As for the range of validity, answer these questions: 1) What is the value of the integral $\int{t^{-n} \, dt}$ for $n = 1$? 2) When does the integrated out part $\frac{x}{2(n - 1)(1 + x^2)^{n - 1}}$ converge when $x \rightarrow \infty$?
P: 643
 Quote by Dickfore Oh, I see you already did the steps. As for the range of validity, answer these questions: 1) What is the value of the integral $\int{t^{-n} \, dt}$ for $n = 1$? 2) When does the integrated out part $\frac{x}{2(n - 1)(1 + x^2)^{n - 1}$ converge when $x \rightarrow \infty$?
 Quote by Dickfore Oh, I see you already did the steps. As for the range of validity, answer these questions: 1) What is the value of the integral $\int{t^{-n} \, dt}$ for $n = 1$? 2) When does the integrated out part $\frac{x}{2(n - 1)(1 + x^2)^{n - 1}}$ converge when $x \rightarrow \infty$?
 P: 3,014 what is $u v$ in the integration by parts?