Completing the square Evaluate the following integrals.

$\int \frac{dx}{{x}^{2}+6x+18}$

a2linetagadaW
2021-10-27
Answered

Completing the square Evaluate the following integrals.

$\int \frac{dx}{{x}^{2}+6x+18}$

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Szeteib

Answered 2021-10-28
Author has **102** answers

Step 1

The given integral is$\int \frac{dx}{{x}^{2}+6x+18}$ .

Evaluate the above integral as follows.

Step 2

$\int \frac{dx}{{x}^{2}+6x+18}=\int \frac{dx}{{(x+3)}^{2}-{3}^{2}+18}$

$=\int \frac{dx}{{(x+3)}^{2}-9+18}$

$=\int \frac{dx}{{(x+3)}^{2}+9}$

$=\int \frac{dx}{{(x+3)}^{2}+{3}^{2}}$

$=\frac{1}{3}{\mathrm{tan}}^{-1}\left(\frac{x+3}{3}\right)+C$

Thus,$\int \frac{dx}{{x}^{2}+6x+18}=\frac{1}{3}{\mathrm{tan}}^{-1}\left(\frac{x+3}{3}\right)+C$ .

The given integral is

Evaluate the above integral as follows.

Step 2

Thus,

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I attempted the integral test but on the interval $[1,\sqrt{2})$ it is increasing and decreasing on $(\sqrt{2},\mathrm{\infty})$. So the integral test in only applicable for the decreasing part. + the integral computation seems to lead to 3 pages of steps...

I believe the comparison test would be the most reasonable test, graphically I observed that ${a}_{n}$ behaves like ${b}_{n}=1/n$ when n is large.

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I attempted the limit comparison, and ratio test, but inconclusive. I am uncertain if I am doing them properly.

How could I bound below ${a}_{n}$ to proceed with the comparison test? Is there a more appropriate method? How would you proceed?

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