$$$- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}$$$ 关于$$$x$$$的积分

求$$$\int \left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)\, dx$$$。

对 $$$c=- 9 i n t \sec{\left(2 \right)}$$$ 和 $$$f{\left(x \right)} = x \sin{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)d x}}} = {\color{red}{\left(- 9 i n t \sec{\left(2 \right)} \int{x \sin{\left(3 x \right)} d x}\right)}}$$

对于积分$$$\int{x \sin{\left(3 x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=x$$$ 和 $$$\operatorname{dv}=\sin{\left(3 x \right)} dx$$$。

则 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\sin{\left(3 x \right)} d x}=- \frac{\cos{\left(3 x \right)}}{3}$$$ (步骤见 »)。

该积分可以改写为

$$- 9 i n t \sec{\left(2 \right)} {\color{red}{\int{x \sin{\left(3 x \right)} d x}}}=- 9 i n t \sec{\left(2 \right)} {\color{red}{\left(x \cdot \left(- \frac{\cos{\left(3 x \right)}}{3}\right)-\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right) \cdot 1 d x}\right)}}=- 9 i n t \sec{\left(2 \right)} {\color{red}{\left(- \frac{x \cos{\left(3 x \right)}}{3} - \int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}\right)}}$$

对 $$$c=- \frac{1}{3}$$$ 和 $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} - {\color{red}{\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}}}\right) = - 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} - {\color{red}{\left(- \frac{\int{\cos{\left(3 x \right)} d x}}{3}\right)}}\right)$$

设$$$u=3 x$$$。

则$$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{3}$$$。

因此，

$$- 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{3}\right) = - 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{3}\right)$$

对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$- 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{3}\right) = - 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{3}\right)}}}{3}\right)$$

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

$$- 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{9}\right) = - 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{{\color{red}{\sin{\left(u \right)}}}}{9}\right)$$

回忆一下 $$$u=3 x$$$:

$$- 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{\sin{\left({\color{red}{u}} \right)}}{9}\right) = - 9 i n t \sec{\left(2 \right)} \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{\sin{\left({\color{red}{\left(3 x\right)}} \right)}}{9}\right)$$

因此，

$$\int{\left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)d x} = - 9 i n t \left(- \frac{x \cos{\left(3 x \right)}}{3} + \frac{\sin{\left(3 x \right)}}{9}\right) \sec{\left(2 \right)}$$

化简：

$$\int{\left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)d x} = i n t \left(3 x \cos{\left(3 x \right)} - \sin{\left(3 x \right)}\right) \sec{\left(2 \right)}$$

加上积分常数：

$$\int{\left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)d x} = i n t \left(3 x \cos{\left(3 x \right)} - \sin{\left(3 x \right)}\right) \sec{\left(2 \right)}+C$$

$$$\int \left(- 9 i n t x \sin{\left(3 x \right)} \sec{\left(2 \right)}\right)\, dx = i n t \left(3 x \cos{\left(3 x \right)} - \sin{\left(3 x \right)}\right) \sec{\left(2 \right)} + C$$$A