Find the definite integral:
$$\int\limits_{-\pi}^{\pi} (x^{199} (\sin x )^{202} + \mid x+2 \mid ^{101}) dx$$.
At first it seemed as a pretty obvious and easy problem and now I have unfortunately struggled with it for two hours and still with no progression. My idea was to split the integral into two halves (one has $-\pi$ as lower and $-2$ as upper bound and the second one $-2$ as lower and $\pi$ as upper) to help to lose the absolute value under it but then I faced the situation where I have to integrate $x^{199} (\sin x )^{202}$ and I have absolutely no clue how to do it.
If someone could explain the problem to me (because I am pretty sure I have the wrong ideas) I would be more than grateful. I have three days until finals and hopefully understanding the problem would help me to pass the exam and help to develop my skills enough to give some of your contribution back to the community in the future.
Best Answer
If $f(x)$ is odd, i.e. $f(x)=-f(-x)$, on $[-a,a]$, $\int_{-a}^{a}f(x)\,dx=0$. That should take care of the first term.