int frac{d x}{x^2+2 x+5} = . . . . . . + C | vedclass

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Question

(B) To evaluate the integral $I = \int \frac{d x}{x^2+2 x+5}$,we first complete the square in the denominator.$x^2 + 2x + 5 = (x^2 + 2x + 1) + 4 = (x+

Vedclass · Accepted Answer

$\frac{1}{2} 	an ^{-1}\left(\frac{x+1}{2}ight)$

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(B) To evaluate the integral $I = \int \frac{d x}{x^2+2 x+5}$,we first complete the square in the denominator.$x^2 + 2x + 5 = (x^2 + 2x + 1) + 4 = (x+1)^2 + 2^2$.Now,the integral becomes $I = \int \frac{d x}{(x+1)^2 + 2^2}$.Using the standard integration formula $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} 	an^{-1}(\frac{x}{a}) + C$,where $a = 2$ and the variable is $(x+1)$:$I = \frac{1}{2} 	an^{-1}\left(\frac{x+1}{2}ight) + C$.Comparing this with the given options,the correct option is $B$.

$\int \frac{d x}{x^2+2 x+5} = $ . . . . . . $+ C$.

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