If int frac{dx}{x^2+2x+2} = f(x) + c, then f( | vedclass

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(A) Let $I = \int \frac{dx}{x^2+2x+2}$.We can rewrite the denominator by completing the square:$x^2 + 2x + 2 = (x^2 + 2x + 1) + 1 = (x+1)^2 + 1$.Subst

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$	an^{-1}(x+1)$

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(A) Let $I = \int \frac{dx}{x^2+2x+2}$.We can rewrite the denominator by completing the square:$x^2 + 2x + 2 = (x^2 + 2x + 1) + 1 = (x+1)^2 + 1$.Substituting this into the integral, we get:$I = \int \frac{dx}{(x+1)^2 + 1}$.Using the standard integral formula $\int \frac{du}{u^2 + a^2} = \frac{1}{a} 	an^{-1}(\frac{u}{a}) + c$, where $u = x+1$ and $a = 1$:$I = 	an^{-1}(x+1) + c$.Given that $I = f(x) + c$, we conclude that $f(x) = 	an^{-1}(x+1)$.

If $\int \frac{dx}{x^2+2x+2} = f(x) + c$, then $f(x)$ is equal to :

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