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Q.
if $\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6$, then
Integrals
Solution:
$\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6\quad....\left(i\right)$
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 $