Q111 / 160 · TS EAMCET 2018 (05 May Shift 2) · Mathematics

$\int \frac{d x}{\sqrt{(x-1)(x-2)}}=$

  • A
    $\sin ^{-1}(2 x+5)+c$
  • B
    $\sinh ^{-1}(2 x-5)+c$
  • Correct answer
    C
    $\cosh ^{-1}(2 x-3)+c$
  • D
    $\sin ^{-1}(3-2 x)+c$

Solution

Let $$ \begin{aligned} I & =\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\int \frac{d x}{\sqrt{x^2-3 x+2}} \\ & =\int \frac{d x}{\sqrt{x^2-2 \cdot \frac{3}{2} \cdot x+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2+2}} \end{aligned} $$ $$ \begin{aligned} & =\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\frac{1}{4}}}=\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\left(\frac{1}{2}\right)^2}} \\ & =\cosh ^{-1}\left(\frac{x-3 / 2}{1 / 2}\right)+c \\ & \quad\left[\because \int \frac{d x}{\sqrt{x^2-a^2}}=\cosh ^{-1}\left(\frac{x}{a}\right)+c\right] \\ & =\cosh ^{-1}(2 x-3)+c \end{aligned} $$

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

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