Q105 / 180 · COMEDK 2017 · Mathematics

If $\int_{a}^{b} \frac{x^{n}}{x^{n}+(16-x)^{n}} d x=6$, then

  • A
    $a=4, b=12, n \in R$
  • Correct answer
    B
    $a=2, b=14, n \in R$
  • C
    $a=-4, b=20, n \in R$
  • D
    $a=2, b=8, n \in R$

Solution

$\int_{a}^{b} \frac{x^{n}}{x^{n}+(16-x)^{n}} d x=6...(i)$ Let $a+b=16$, then $\int_{a}^{b} \frac{(16-x)^{n}}{(16-x)^{n}+x^{n}} d x=6..(ii)$ Adding Eqs. (i) and (ii), we get $\int_{a}^{b} 1 \cdot d x=12 \Rightarrow b-a=12$ Solving $a+b=16$ and $b-a=12$, we get $a=2, b=14, n \in R$

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

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