Q116 / 160 · AP EAMCET 2022 (08 Jul Shift 1) · Mathematics

$\int_2^5 \sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}} d x=$

  • A
    $16 / 3$
  • B
    $32 / 3$
  • Correct answer
    C
    $28 / 3$
  • D
    $4 / 3$

Solution

$I=\int_q^5(\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}) d x$ $\begin{aligned} & \int_{\mathrm{q}}^5\left(\sqrt{(\mathrm{x}-1)^2+2 \sqrt{\mathrm{x}-1}+1}+\sqrt{(\mathrm{x}-1)^2-2 \sqrt{\mathrm{x}-1+1}}\right) \mathrm{dx} \\ & \int_{\mathrm{q}}^5[(\sqrt{\mathrm{x}-1}+1)+(\sqrt{\mathrm{x}-1}-1)] \mathrm{dx}\end{aligned}$ $\begin{aligned} & \int_{\mathrm{q}}^5(\mathrm{x}-1)^{1 / 2} \mathrm{dx}=\mathrm{q} \times \frac{2}{3}\left[(\mathrm{x}-1)^{3 / 2}\right]_2^5 \\ & \frac{4}{3} \times[8-1]=\frac{28}{3}\end{aligned}$

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

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