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# Choose the correct answer in $\Large \int \normalsize\frac{\large dx}{\large x^2+2x+2}$ equals

$\begin{array}{1 1} (A)\;x\tan^{-1}(x+1)+C \\ (B)\;\tan^{-1}(x+1)+C \\ (C)\;(x+1)\tan^{-1}x+C \\ (D)\;\tan^{-1}x+C \end{array}$

Toolbox:
• $\int\frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\big(\frac{x}{a}\big)+c.$
Given $I=\int\frac{dx}{x^2+2x+2}.$

$x^2+2x+2$ can be written as $(x+1)^2+1^2.$

Therefore $I=\int\frac{dx}{(x+1)^2+1^2}$.

On integrating we get,

$\;\;\;=\tan^{-1}(x+1)+c.$

Hence the correct answer is B.