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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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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