Question

If $\int \frac{dx}{x^2+2x+2}=f\left(x\right)+C$, then $f\left(x\right)=$

• A. ${\tan }^{-1}\left(x+1\right)$
• B. $2{\tan }^{-1}\left(x+1\right)$
• C. $-{\tan }^{-1}\left(x+1\right)$
• D. $3{\tan }^{-1}\left(x+1\right)$

Answer 1

The correct option is A

${\tan }^{-1}\left(x+1\right)$

Explanation for the correct option.

Find the function $f\left(x\right)$.

The integral $\int \frac{dx}{x^2+2x+2}$ can be solved as:

$\begin{array}{rcl}\int \frac{dx}{x^2+2x+2}& =& \int \frac{dx}{x^2+2x+1+1}\\ & =& \int \frac{dx}{{\left(x+1\right)}^2+1}\\ & =& {\tan }^{-1}\left(x+1\right)+C\left[\frac{d}{dx}\left({\tan }^{-1}x\right)=\frac{1}{1+x^2}\right]\end{array}$

As it is given that $\int \frac{dx}{x^2+2x+2}=f\left(x\right)+C$, so the function $f\left(x\right)$ is equal to ${\tan }^{-1}\left(x+1\right)$.

Hence the correct option is A.