Q115 / 160 · AP EAMCET 2022 (05 Jul Shift 1) · Mathematics

$\int_1^3 x^n \sqrt{x^2-1} d x=6$, then $n=$

  • A
    $2$
  • Correct answer
    B
    $3$
  • C
    $4$
  • D
    $5$

Solution

Let $I=\int_1^3 x \sqrt[n]{x^2-1} d x=6$ $=\frac{1}{2} \int_1^3 2 x\left(x^2-1\right)^{\frac{1}{n}} d x=6$ $\Rightarrow\left[\frac{\left(n^2-1\right)^{\frac{1}{n}+1}}{\frac{1}{n}+1}\right]_1^3=12$ $\Rightarrow \quad \frac{(8)^{\frac{1}{n}+1}}{\frac{1}{n}+1}=12$ $\Rightarrow \quad \frac{(8)^{\frac{1}{n}}}{(n+1)}(n)=\frac{3}{2}$ Now, on putting $n=3$, L H S $=$ R. H S $\therefore n=3$

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

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