$\int_{-\pi / 4}^{7 \pi / 4}(\sin x+\cos x) d x=\square$

Step 1 ：Given the integral \(\int_{-\pi / 4}^{7 \pi / 4}(\sin x+\cos x) d x\), we need to find its value.

Step 2 ：The first step is to find the antiderivative of the function \(\sin(x) + \cos(x)\). The antiderivative of \(\sin(x)\) is \(-\cos(x)\), and the antiderivative of \(\cos(x)\) is \(\sin(x)\). So, the antiderivative of \(\sin(x) + \cos(x)\) is \(-\cos(x) + \sin(x)\).

Step 3 ：Next, we apply the Fundamental Theorem of Calculus, which states that the definite integral of a function from a to b is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a.

Step 4 ：We evaluate the antiderivative at \(7\pi/4\) and \(-\pi/4\) and subtract the two results to find the definite integral. The result is 0.

Step 5 ：This means that the area under the curve of the function from \(-\pi/4\) to \(7\pi/4\) is 0.

Step 6 ：Final Answer: \(\boxed{0}\)