$\text{A.}$ $F(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}-x\right), x \leq 0 \\ (x+1) \cos x-\sin x, x > 0\end{array}\right.$ $\text{B.}$ $F(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}-x\right)+1, x \leq 0 \\ (x+1) \cos x-\sin x, x > 0\end{array}\right.$ $\text{C.}$ $F(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}-x\right), x \leq 0 \\ (x+1) \sin x+\cos x, x > 0\end{array}\right.$ $\text{D.}$ $F(x)=\left\{\begin{array}{l}\ln \left(\sqrt{1+x^2}+x\right)+1, x \leq 0 \\ (x+1) \sin x+\cos x, x > 0\end{array}\right.$
【答案】 D

【解析】 当 $x \leq 0$ 时,
$$\int f(x) d x=\int \frac{d x}{\sqrt{1+x^2}}=\ln \left(x+\sqrt{1+x^2}\right)+C_1$$

\begin{aligned} & \int f(x) d x=\int(x+1) \cos x d x=\int(x+1) d \sin x=(x+1) \sin x-\int \sin x d x \\ & =(x+1) \sin x+\cos x+C_2 \end{aligned}

$$\lim _{x \rightarrow 0^{-}} \ln \left(x+\sqrt{1+x^2}\right)+C_1=C_1, \lim _{x \rightarrow 0^{+}}(x+1) \sin x+\cos x+C_2=1+C_2$$

$$\int f(x) d x=\left\{\begin{array}{l} \ln \left(x+\sqrt{1+x^2}\right)+1+C, x \leq 0 \\ (x+1) \sin x+\cos x+C, x > 0 \end{array},\right.$$