Q110 / 180 · KCET 2010 · Mathematics

If $\int \frac{\sqrt{x}}{x(x+1)} d x=k \tan ^{-1} m$, then $(k, m)$ is

  • A
    $(2, \mathrm{x})$
  • B
    $(1, \mathrm{x})$
  • C
    $(1, \sqrt{\mathrm{x}})$
  • Correct answer
    D
    $(2, \sqrt{\mathrm{x}})$

Solution

$\begin{aligned} &\int \frac{\sqrt{x}}{x(x+1)} d x=k \tan ^{-1} m \\ &\text { Put } \quad\left\{\begin{array}{l} x=\tan ^{2} \theta \\ d x=2 \tan \theta \cdot \sec ^{2} \theta d \theta \end{array}\right. \\ &\quad=\int \frac{\tan \theta}{\tan ^{2} \theta \cdot \sec ^{2} \theta} \cdot\left(2 \tan \theta \cdot \sec ^{2} \theta\right) d \theta \\ &=2 \int d \theta=2 \theta=2 \tan ^{-1} \sqrt{x}=k \tan ^{-1}(m) \end{aligned}$ On comparing, we get $(\mathrm{k}, \mathrm{m})=(2, \sqrt{\mathrm{x}})$

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

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