$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to

The function is $(x^7 + x^5 + x^3 + x + 1)$ with limits from $-1$ to $1$ (symmetric around zero).

The function can be split into:

• Odd power terms: $x^7, x^5, x^3, x$
• Even power term: $1$

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When integrating an odd function over symmetric limits (like $-1$ to $1$), the result is always zero because the negative side perfectly cancels the positive side.

$\int_{-1}^{1}x^7 dx = 0$

$\int_{-1}^{1}x^5 dx = 0$

$\int_{-1}^{1}x^3 dx = 0$

$\int_{-1}^{1}x dx = 0$

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The constant term remains:

$\int_{-1}^{1}1 \, dx = [x]_{-1}^{1}$

$= (1) - (-1)$

$= 2$

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$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx = 0 + 0 + 0 + 0 + 2$

$= 2$