Q82 / 170 · Manipal (MET) 2017 · Mathematics

$\int \frac{d x}{x^2+4 x+13}$ is equal to

  • A
    $\log \left(x^2+4 x+130\right)+c$
  • Correct answer
    B
    $\frac{1}{3} \tan ^{-1}\left(\frac{x+2}{3}\right)+c$
  • C
    $\log (2 x+4)+c$
  • D
    $\frac{1}{x^2+4 x+13}+c$

Solution

$\begin{aligned} & \text { Let } I=\int \frac{d x}{x^2+4 x+13} \\ & =\int \frac{d x}{x^2+4 x+4+3^2}=\int \frac{d x}{\left(x+2)^2+3^2\right.} \\ & =\frac{1}{3} \tan ^{-1}\left(\frac{x+2}{3}\right)+c\end{aligned}$

Hence, the correct answer is option: 'B'.

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