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

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

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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 $\int \frac{du}{u^2+1} = 	an^{-1}(u) + c$,where $u = x+1$ and $du = dx$:$I = 	an^{-1}(x+1) + c$.Comparing this with $I = f(x) + c$,we find 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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