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

The correct answer is Option (3) → 2

The terms \(x^7, x^5, x^3, x\) are odd functions, and \(\int_{-a}^{a} f(x)\,dx = 0\) for an odd f:

    \( \displaystyle \int_{-1}^{1} (x^7+x^5+x^3+x)\,dx = 0 \)

Only the constant term is left:

    \( \displaystyle \int_{-1}^{1} 1\,dx = \Big[x\Big]_{-1}^{1} = 1 - (-1) = 2 \)

So the integral equals 2.