Question 1 (1 point) Let $f(x)=\left\{\begin{array}{ll}2 x^{4}, & -\pi \leq x<0 \\ 5 \sin (x), & 0 \leq x \leq \pi\end{array}\right.$. Evaluate the definite integral $\int_{-\pi}^{\pi} f(x) d x$ $\frac{4}{5} \pi^{5}-10$ $8 \pi^{3}-10$ $\frac{2}{5} \pi^{5}+10$ $\frac{4}{5} \pi^{5}$ $\frac{2}{5} \pi^{5}$

Step 1 ：The given function is a piecewise function, so we will need to calculate the definite integral separately for each piece and then add the results together.

Step 2 ：The antiderivative of \(2x^4\) is \(\frac{2}{5}x^5\) and the antiderivative of \(5\sin(x)\) is \(-5\cos(x)\).

Step 3 ：We will need to evaluate these at the appropriate points and subtract to find the definite integral.

Step 4 ：The definite integral of the function from \(-\pi\) to \(0\) is \(\frac{2}{5}\pi^5\).

Step 5 ：The definite integral of the function from \(0\) to \(\pi\) is \(10\).

Step 6 ：The definite integral of the function from \(-\pi\) to \(\pi\) is the sum of the definite integrals of the two pieces of the function, which we calculated to be \(10 + \frac{2}{5}\pi^5\).

Step 7 ：Final Answer: \(\boxed{\frac{2}{5} \pi^{5}+10}\)