Q111 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

$\int \frac{\mathrm{x}^{9 / 2}}{\sqrt{1+\mathrm{x}^{11}}} \mathrm{dx}$ is -

  • Correct answer
    A
    $\frac{2}{11} \log \left(\mathrm{x}^{7 / 2}+\sqrt{\mathrm{x}^7+1}\right)+\mathrm{c}$
  • B
    $\frac{1}{2} \log \frac{\mathrm{x}^{11}+1}{\mathrm{x}^{11}-1}+\mathrm{c}$
  • C
    $2 \sqrt{1+\mathrm{X}^{11}}+\mathrm{c}$
  • D
    None of these

Solution

$$ \begin{aligned} & \text { Let } \mathrm{x}^{11 / 2}=\mathrm{t} \Rightarrow \frac{11}{2} \mathrm{x}^{9 / 2} \mathrm{dx}=\mathrm{dt} \\ & \Rightarrow \mathrm{I}=\frac{2}{11} \int \frac{\mathrm{dt}}{\sqrt{1+\mathrm{t}^2}}=\frac{2}{11} \log \left(\mathrm{t}+\sqrt{1+\mathrm{t}^2}\right)+\mathrm{c} \\ & =\frac{2}{11} \log \left(\mathrm{x}^{11 / 2}+\sqrt{1+\mathrm{x}^{11}}\right)+\mathrm{c} . \end{aligned} $$

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

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