TITLE: Use properties of the integral and calculate, without extensive accounts:
QUESTION [1 upvotes]: $$\int_{-5}^5 x^2-x^{10}\sin(x)+x^5\sqrt{1+x^4}$$
Which properties can i use to solve this quickly?

REPLY [2 votes]: If $f(x)$ is an odd function then $\int_{-a}^af(x)\,dx=?$

REPLY [2 votes]: Hint: the integral of an odd function on an interval symmetric about the origin, is $0$. Or symbolically, if $f$ is odd ($f(-x)=-f(x)$), you have:
$$\color{blue}{\int_{-a}^a f(x) \,\mbox{d}x = 0}$$
Combine this with the linearity of the integral:
$$\int_{-5}^5 x^2-x^{10}\sin(x)+x^5\sqrt{1+x^4} \,\mbox{d}x \\=
\int_{-5}^5 x^2 \,\mbox{d}x
-
\color{blue}{\int_{-5}^5 x^{10}\sin(x) \,\mbox{d}x}
+
\color{blue}{\int_{-5}^5 x^5\sqrt{1+x^4} \,\mbox{d}x}$$

REPLY [2 votes]: Split interval using additivity.
The first is even, the other two are odds.
Second and third integrals are equal to $0$ as other answers have already said, the first is obvious.