Question

Let \([\mathrm{t}]\) denote the greatest integer less than or equal to t. Then the value of \(10-10 \int_{-2}^{2}\left[x+x^{3}\right] dx\) is ________.

Hint:

Let \(g(x)=x+x^3\), an odd function; for non-integer t we have \([t]+[-t]=-1\), so pair x with -x.

Answer 1

Correct Answer: 30

Let \(g(x)=x+x^3\). It is odd, so \(g(-x)=-g(x)\), and it is strictly increasing, so g(x) is an integer only at finitely many points (which do not affect the integral).
Let \(I=\int_{-2}^{2}[g(x)]\,dx\). Replacing x by -x gives \(I=\int_{-2}^{2}[g(-x)]\,dx=\int_{-2}^{2}[-g(x)]\,dx\).
Adding: \(2I=\int_{-2}^{2}\big([g(x)]+[-g(x)]\big)dx\).
For non-integer t, \([t]+[-t]=-1\). Therefore \(2I=\int_{-2}^{2}(-1)\,dx=-4\), so \(I=-2\).
Required value \(=10-10(-2)=30\).
Answer: 30