Improper Integral: Solving with Substitution Method

[tahayassen]

$$\begin{array}{l} \int\limits_{ - \infty }^\infty {\frac{{{x^2}}}{{{x^6} + 9}}} \\ = \int\limits_{ - \infty }^0 {\frac{{{x^2}}}{{{x^6} + 9}}} + \int\limits_0^\infty {\frac{{{x^2}}}{{{x^6} + 9}}} \\ = \mathop {\lim }\limits_{t \to - \infty } \int\limits_t^0 {\frac{{{x^2}}}{{{x^6} + 9}}} + \mathop {\lim }\limits_{m \to \infty } \int\limits_0^m {\frac{{{x^2}}}{{{x^6} + 9}}} \end{array}$$

I had this problem on a test yesterday and I couldn't solve it. I tried to reduce the denominator to irreducible quadratic factors and then use partial fractions, but it was impossible. I couldn't see an obvious substitution, but apparently you were supposed to solve it by substitution.

 

[danielakkerma]

Have you tried the following?
$$x^6+9 = (x^2+\sqrt[3]{3})(x^4-x^2\sqrt[3]{3}+3\sqrt[3]{3})$$?
Then via Fractional decomposition?
(Edited!, mistook +9, for -9)

 

[tahayassen]

Have you tried the following?
$$x^6+9 = (x^3-3)(x^3+3)$$?
Then:
$$\frac{1}{(x^3-3)(x^3+3)} = \frac{1}{2}(\frac{1}{x^3-3}+\frac{1}{x^3+3})$$

HOW DID YOU GET THOSE FACTORS? I spent ages trying to factor it!

Edit: Wait a second. You can't apply partial fractions to cubic factors.

 

[danielakkerma]

Yes, sorry, I edited my original post :)...
Spoke too soon. How about the alternative?

 

[danielakkerma]

Okay, here's a way:
Try dividing the denominator by 9;
You should get:
$$\frac{1}{9}\frac{x^2}{\frac{x^6}{9}+1}$$
Then set z^2(!) = x^6/9;
By differentiation:
$$2zdz=\frac{6x^5}{9}dx$$
Substitute, and integrate as necessary.

 

[HallsofIvy]

$(x^3+ 3)(x^3- 3)= x^6- 9$, not $x^6+ 9$, which is why daniel akkerma changed his factorization. (But too late, his mistake will "live in infamy"!):tongue:

Personally, I see no reason to factor $x^6+ 9$ at all. Instead, with that "$x^2$" in the numerator, let $u= x^3$ so that $du= 3x^2dx$ and $(1/3)du= x^2dx$ so the integral becomes
$$\frac{1}{3}\int\frac{du}{u^2+ 6}$$
which can be integrated as an arctangent.

(edit- That appears to be what daniel akkerma is suggesting the his post just before mine- that was posted while I was still typing.)

 

[danielakkerma]

Yeah, I am really sorry about, Ivy & Taha.
For some reason I descried +9 as -9... Mysterious are the ways of the brain, etc, etc, etc :D...
Sorry!
Excellent choice of substitution from you, btw. I still like mine via z^2 though; Feels more direct.

 

[Mark44]

One thing that no one mentioned yet is that since the integrand is an even function, it suffices to find $\int_0^{\infty} f(x) dx$ and double it.