# Completing the square Evaluate the following integrals. \int \frac{dx}{x^{2}+6x+18}

Completing the square Evaluate the following integrals.
$\int \frac{dx}{{x}^{2}+6x+18}$
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Step 1
The given integral is $\int \frac{dx}{{x}^{2}+6x+18}$.
Evaluate the above integral as follows.
Step 2
$\int \frac{dx}{{x}^{2}+6x+18}=\int \frac{dx}{{\left(x+3\right)}^{2}-{3}^{2}+18}$
$=\int \frac{dx}{{\left(x+3\right)}^{2}-9+18}$
$=\int \frac{dx}{{\left(x+3\right)}^{2}+9}$
$=\int \frac{dx}{{\left(x+3\right)}^{2}+{3}^{2}}$
$=\frac{1}{3}{\mathrm{tan}}^{-1}\left(\frac{x+3}{3}\right)+C$
Thus, $\int \frac{dx}{{x}^{2}+6x+18}=\frac{1}{3}{\mathrm{tan}}^{-1}\left(\frac{x+3}{3}\right)+C$.