Question

Integration Help...

Answer 1

What is k if \[\int\limits_{-2}^{2} (x^7 + k) dx = 16\]

Answer 2

Try integrating \(x^7\) on that interval.

Alternatively, take a look at this:

If \(f(-x)=-f(x)\), then \(\int_{-a}^{a}f(x)dx=0. \)

It's a really neat fact because you can prove it very easily (it's a good exercise) and a lot of functions have the required property [They're called odd functions and notably include sine, tangent, and odd-order polynomials whose graphs pass through the origin].

\(p(x)=x^7\) is one of these functions, and you can save yourself an integral by noticing that. Integrals like this turn up quite often, so this result is more helpful than it might appear at first.

Answer 3

So the answer is 4. Thank you.