Question:
$\int \frac{d x}{x^{2}+2 x+2}$ equals
A. $x \tan ^{-1}(x+1)+C$
B. $\tan ^{-1}(x+1)+C$
C. $(x+1) \tan ^{-1} x+\mathrm{C}$
D. $\tan ^{-1} x+\mathrm{C}$
Solution:
$\int \frac{d x}{x^{2}+2 x+2}=\int \frac{d x}{\left(x^{2}+2 x+1\right)+1}$
$=\int \frac{1}{(x+1)^{2}+(1)^{2}} d x$
$=\left[\tan ^{-1}(x+1)\right]+\mathrm{C}$
Hence, the correct answer is B.
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