Question

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

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