$$$2 x \cos{\left(x^{2} \right)}$$$ 的积分

求$$$\int 2 x \cos{\left(x^{2} \right)}\, dx$$$。

设$$$u=x^{2}$$$。

则$$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$ (步骤见»)，并有$$$x dx = \frac{du}{2}$$$。

因此，

$${\color{red}{\int{2 x \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\int{\cos{\left(u \right)} d u}}}$$

余弦函数的积分为 $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$：

$${\color{red}{\int{\cos{\left(u \right)} d u}}} = {\color{red}{\sin{\left(u \right)}}}$$

回忆一下 $$$u=x^{2}$$$:

$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{x^{2}}} \right)}$$

因此，

$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}$$

加上积分常数：

$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}+C$$

$$$\int 2 x \cos{\left(x^{2} \right)}\, dx = \sin{\left(x^{2} \right)} + C$$$A