Convergence or Divergence integral

[kilroymcb]

I have the definite integral from 1 to infinity of 9/4x^2 + 22x + 10. The problem asks me to find if this diverges or converges and if so, to evaluate.

Now, first I turn the integral into a lim from 1 to N. Then, I treat the problem as an indefinite integral and try and bust out something I can work with

Substitution won't work. I can't see any obvious trig subs. Partial fractions will not work, nor will long division. Soooo, I decide to complete the square of the denominator to see if I can get something that looks workable.

First, of course, I move the 9 outside the integral. Then, 4(x^2+ 22/4 + 10/4). I take the 4 out and put it beneath the 9 for 9/4 int 1/(x^2 + 22/4 + 10/4). Now, I complete the square 1/(x^2 + 22/4 + 11/4) + 10/4 - 11/4 for 1/(x+11/4)^2 - 1/4...

The obvious thing at this point is to do partial fractions. But I'm not feeling good about the way this is going. Does anyone have any suggestions?

 

[pka]

\(\displaystyle \L\begin{array}{l}
x \ge 1,\quad \Rightarrow \quad 4x^2 + 22x + 10 > x^2 \\
\quad \Rightarrow \quad \frac{9}{{4x^2 + 22x + 10}} < \frac{9}{{x^2 }} \\
\quad \quad \\
\quad \Rightarrow \quad \int\limits_1^\infty {\frac{{9dx}}{{4x^2 + 22x + 10}}} < \int\limits_1^\infty {\frac{{9dx}}{{x^2 }}} \\
\end{array}\)

 

[kilroymcb]

\(\displaystyle \L\begin{array}{l}
x \ge 1,\quad \Rightarrow \quad 4x^2 + 22x + 10 > x^2 \\
\quad \Rightarrow \quad \frac{9}{{4x^2 + 22x + 10}} < \frac{9}{{x^2 }} \\
\quad \quad \\
\quad \Rightarrow \quad \int\limits_1^\infty {\frac{{9dx}}{{4x^2 + 22x + 10}}} < \int\limits_1^\infty {\frac{{9dx}}{{x^2 }}} \\
\end{array}\)

So, you're saying that taking the limit of 9dx/x^2 will tell me whether 9/4x^2 + 22x + 10 converges or diverges?

 

[pka]

Yes, you have that right.

 

[kilroymcb]

Yes, you have that right.

Excellent! So, Int from 1 to N of 9/x^2 = -9/N + 9. Which = -9/infinity which goes to zero and therefore the limit converges at 9?

 

[pka]

Int from 1 to N of 9/x^2 = -9/N + 9. Which = -9/infinity which goes to zero and therefore the limit converges at 9?

Yes the integral converges to some number < 9.

 

[kilroymcb]

Int from 1 to N of 9/x^2 = -9/N + 9. Which = -9/infinity which goes to zero and therefore the limit converges at 9?

Yes the integral converges to some number < 9.

Good... Now, is it correct to do the integral of 1/4x^2 + 22x + 10 by completing the square of the denominator then using partial fractions?