Q114 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

$\int \frac{d x}{(x-2) \sqrt{x^2-3 x+5}}=$

  • A
    $\frac{-1}{\sqrt{3}} \cosh ^{-1}\left[\frac{7 x-8}{\sqrt{37}(x-2)}\right]+C$
  • Correct answer
    B
    $\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{x+4}{\sqrt{11}(x-2)}\right]+C$
  • C
    $\frac{-1}{\sqrt{3}} \cosh ^{-1}\left[\frac{x+4}{\sqrt{11}(x-2)}\right]+C$
  • D
    $\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{7 x-8}{\sqrt{37}(x-2)}\right]+C$

Solution

Let $I=\int \frac{d x}{(x-2) \sqrt{x^2-3 x+5}}$ Put, $x-2=\frac{1}{t} d x=\frac{-1}{t^2} d t \Rightarrow x=2+\frac{1}{t}$ $\begin{aligned} \therefore \quad I & =\int_{\frac{1}{t}}^{-1 / t^2} \frac{d t}{\sqrt{\left(2+\frac{1}{t}\right)^2-3\left(2+\frac{1}{t}\right)+5}} \\ I & =\int-\frac{d t}{\sqrt{3 t^2+t+1}}=\frac{-1}{\sqrt{3}} \int \frac{d t}{\left(t+\frac{1}{6}\right)^2+\frac{\sqrt{11}}{6}} \\ I & =\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{6 t+1}{\sqrt{11}}\right] \\ I & =\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{6+(x-2)}{\sqrt{11}(x-2)}\right]+C\left[t=\frac{1}{x-2}\right] \\ I & =\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{x+4}{\sqrt{11}(x-2)}\right]+C \end{aligned}$

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

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