∫limits_-1^1(x^7+x^5+x^3+x+1)dx is equal to

The correct answer is Option (3) → 2

$\displaystyle \int_{-1}^{1}(x^{7}+x^{5}+x^{3}+x+1)\,dx$

Odd functions integrate to $0$ over $[-1,1]$:

$\displaystyle \int_{-1}^{1}x^{7}dx=0,\;\int_{-1}^{1}x^{5}dx=0,\;\int_{-1}^{1}x^{3}dx=0,\;\int_{-1}^{1}x\,dx=0$

Only the constant term contributes:

$\displaystyle \int_{-1}^{1}1\,dx=2$

The value of the integral is $2$.