Mathematics

$$\int \frac{x+2}{x^{2}+2x+5}$$ is equal to


SOLUTION
$$\displaystyle\int \dfrac{x+2}{x^2+2x+5}$$
$$x+2=\lambda\dfrac{d}{dx}(x^2+2x+5)+\mu$$
$$x+2=2\lambda x+2\lambda +\mu$$
$$2\lambda =1$$, $$2\lambda +\mu =2$$
$$\lambda =\dfrac{1}{2}$$, $$\mu =1$$
$$\dfrac{x+2}{x^2+2x+5}=\dfrac{1}{2}\dfrac{(2x+2)}{x^2+2x+5}+\dfrac{1}{x^2+2x+5}$$
$$\displaystyle\int \dfrac{x+2}{x^2+2x+5}=\dfrac{1}{2}log|x^2+2x+5|+\displaystyle\int \dfrac{1}{x^2+2x+5}$$
$$=\dfrac{1}{2}log|x^2+2x+5|+\displaystyle\int \dfrac{1}{(x+1)^2+(2)^2}$$
$$=\dfrac{1}{2} log |x^2+2x+5|+\dfrac{1}{2}\tan^{-1}\left(\dfrac{x+1}{2}\right)$$.
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