Question

if $\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6$, then

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

Answer 1

Answer: (B)

$a = 2, b = 14, n\in R$

Answer 2

$\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6\quad....\left(i\right)$ Let $a + b = 16$, then by property $\int\limits_{a}^{b} \frac{\left(16-x\right)^{n}}{\left(16-x\right)^{n }+ x^{n}} dx = 6\quad....\left(ii\right)$ Adding$\left(i\right)\,\,$ and $\,\left(ii\right)$, we get $\int\limits_{a}^{b} 1\cdot dx = 12$ $\Rightarrow b - a = 12$ Solving $a + b = 16$ and $b- a = 12$, we get $a = 2, b = 14$ and $n\in R$