Q114 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

$\int \frac{2 x^2-1+x^2 \sqrt{x^2+4}}{x^2\left(x^2+4\right)} d x=$

  • A
    $\frac{9}{8} \tan ^{-1} \frac{x}{2}+\frac{1}{4 x}+\cosh ^{-1} \frac{x}{2}+c$
  • Correct answer
    B
    $\frac{9}{8} \tan ^{-1} \frac{x}{2}+\frac{1}{4 x}+\sinh ^{-1} \frac{x}{2}+c$
  • C
    $\frac{9}{16} \log \left|\frac{x+2}{x-2}\right|+\frac{1}{4 x}+\log \left|\frac{x+\sqrt{x^2+4}}{2}\right|+c$
  • D
    $\frac{9}{16} \log \left|\frac{2-x}{2+x}\right|+\frac{1}{4 x}+\cosh ^{-1} \frac{x}{2}+c$

Solution

$\begin{aligned} & \int \frac{2 x^2-1+x^2 \sqrt{x^2+4}}{x^2\left(x^2+4\right)} d x=\frac{9}{4} \\ & \Rightarrow \int \frac{d x}{x^2+4}-\frac{1}{4} \int \frac{d x}{x^2}+\int \frac{d x}{\sqrt{x^2+4}} \\ & \quad=\frac{9}{8} \tan ^{-1}\left(\frac{x}{2}\right)+\frac{1}{4 x}+\sinh ^{-1}\left(\frac{x}{2}\right)+c .\end{aligned}$

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

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