A Tricky Integral for some

Ishan Dasgupta Samarendra here is the solution

using the property $f(x)=f(2a-x)$ the limit becomes half

$2 \displaystyle \int^{\frac{1}{2}}_{0} (x^{n}+(1-x)^{n})^{\frac{1}{n}} dx$ ..... $(1)$

plz see carefully.

for $x\in (0,\frac{1}{2})$

$x$ has smaller value than $(1-x)$ .

taking common $(1-x)^{n}$ in $(1)$

also $\displaystyle \lim_{n\to \infty} (\frac{x}{1-x})^{n} \rightarrow 0$

the integral reduces to

$2 \displaystyle \int^{\frac{1}{2}}_{0} (1-x) \times (1^{n})^{\frac{1}{n}}.dx$

$2 \displaystyle \int^{\frac{1}{2}}_{0} (1-x).dx$

$\frac{3}{4}$

hope it helps