Is the Reduction Formula Valid for All n in Integral Calculations?

[jimbobian]

Homework Statement

If I_n 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.

Homework Equations

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

 

[andrien]

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<n<1/2, I(n+1) is negative.

 

[Dickfore]

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.

 

[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 $x \rightarrow \infty$?

 

[Whovian]

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 $x \rightarrow \infty$?

Please preview your posts, LaTeX errors happen all the time.

 

[jimbobian]

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 $x \rightarrow \infty$?

Thanks for your response.

1) ln(t), so this would suggest it doesn't converge for n=1?

2) Firstly I'm not sure where you've got this fraction from, I can't find it in any working of yours or mine? It would converge for n>1 for sure. For n=1 it won't converge, but for n<1 I have no idea?

 

[Dickfore]

what is $u v$ in the integration by parts?