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

求$$$\int t \sin{\left(t^{2} \right)} \cos{\left(t^{2} \right)}\, dt$$$。

设$$$u=t^{2}$$$。

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

积分变为

$${\color{red}{\int{t \sin{\left(t^{2} \right)} \cos{\left(t^{2} \right)} d t}}} = {\color{red}{\int{\frac{\sin{\left(2 u \right)}}{4} d u}}}$$

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(u \right)} = \sin{\left(2 u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{\sin{\left(2 u \right)}}{4} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(2 u \right)} d u}}{4}\right)}}$$

设$$$v=2 u$$$。

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

该积分可以改写为

$$\frac{{\color{red}{\int{\sin{\left(2 u \right)} d u}}}}{4} = \frac{{\color{red}{\int{\frac{\sin{\left(v \right)}}{2} d v}}}}{4}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(v \right)} = \sin{\left(v \right)}$$$ 应用常数倍法则 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$：

$$\frac{{\color{red}{\int{\frac{\sin{\left(v \right)}}{2} d v}}}}{4} = \frac{{\color{red}{\left(\frac{\int{\sin{\left(v \right)} d v}}{2}\right)}}}{4}$$

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

$$\frac{{\color{red}{\int{\sin{\left(v \right)} d v}}}}{8} = \frac{{\color{red}{\left(- \cos{\left(v \right)}\right)}}}{8}$$

回忆一下 $$$v=2 u$$$:

$$- \frac{\cos{\left({\color{red}{v}} \right)}}{8} = - \frac{\cos{\left({\color{red}{\left(2 u\right)}} \right)}}{8}$$

回忆一下 $$$u=t^{2}$$$:

$$- \frac{\cos{\left(2 {\color{red}{u}} \right)}}{8} = - \frac{\cos{\left(2 {\color{red}{t^{2}}} \right)}}{8}$$

因此，

$$\int{t \sin{\left(t^{2} \right)} \cos{\left(t^{2} \right)} d t} = - \frac{\cos{\left(2 t^{2} \right)}}{8}$$

加上积分常数：

$$\int{t \sin{\left(t^{2} \right)} \cos{\left(t^{2} \right)} d t} = - \frac{\cos{\left(2 t^{2} \right)}}{8}+C$$

$$$\int t \sin{\left(t^{2} \right)} \cos{\left(t^{2} \right)}\, dt = - \frac{\cos{\left(2 t^{2} \right)}}{8} + C$$$A