$$$x e^{3} \cos{\left(x e^{3} \right)}$$$ 的積分
您的輸入
求$$$\int x e^{3} \cos{\left(x e^{3} \right)}\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=e^{3}$$$ 與 $$$f{\left(x \right)} = x \cos{\left(x e^{3} \right)}$$$：
$${\color{red}{\int{x e^{3} \cos{\left(x e^{3} \right)} d x}}} = {\color{red}{e^{3} \int{x \cos{\left(x e^{3} \right)} d x}}}$$
對於積分 $$$\int{x \cos{\left(x e^{3} \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。
令 $$$\operatorname{u}=x$$$ 與 $$$\operatorname{dv}=\cos{\left(x e^{3} \right)} dx$$$。
則 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\cos{\left(x e^{3} \right)} d x}=\frac{\sin{\left(x e^{3} \right)}}{e^{3}}$$$（步驟見 »）。
因此，
$$e^{3} {\color{red}{\int{x \cos{\left(x e^{3} \right)} d x}}}=e^{3} {\color{red}{\left(x \cdot \frac{\sin{\left(x e^{3} \right)}}{e^{3}}-\int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} \cdot 1 d x}\right)}}=e^{3} {\color{red}{\left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} d x}\right)}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=e^{-3}$$$ 與 $$$f{\left(x \right)} = \sin{\left(x e^{3} \right)}$$$：
$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - {\color{red}{\int{\frac{\sin{\left(x e^{3} \right)}}{e^{3}} d x}}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - {\color{red}{\frac{\int{\sin{\left(x e^{3} \right)} d x}}{e^{3}}}}\right)$$
令 $$$u=x e^{3}$$$。
則 $$$du=\left(x e^{3}\right)^{\prime }dx = e^{3} dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{e^{3}}$$$。
因此，
$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\sin{\left(x e^{3} \right)} d x}}}}{e^{3}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{e^{3}} d u}}}}{e^{3}}\right)$$
套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=e^{-3}$$$ 與 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$：
$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{e^{3}} d u}}}}{e^{3}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{e^{3}}}}}{e^{3}}\right)$$
正弦函數的積分為 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$：
$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{e^{6}}\right)$$
回顧一下 $$$u=x e^{3}$$$：
$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{u}} \right)}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{x e^{3}}} \right)}}{e^{6}}\right)$$
因此，
$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left(x e^{3} \right)}}{e^{6}}\right) e^{3}$$
化簡：
$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}$$
加上積分常數：
$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}+C$$
答案
$$$\int x e^{3} \cos{\left(x e^{3} \right)}\, dx = \left(x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}\right) + C$$$A