$$$84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$$ 對 $$$x$$$ 的積分

求$$$\int 84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=84 i n t$$$ 與 $$$f{\left(x \right)} = \sin{\left(3 x \right)} \cos{\left(3 x \right)}$$$：

$${\color{red}{\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}} = {\color{red}{\left(84 i n t \int{\sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}\right)}}$$

令 $$$u=\sin{\left(3 x \right)}$$$。

則 $$$du=\left(\sin{\left(3 x \right)}\right)^{\prime }dx = 3 \cos{\left(3 x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(3 x \right)} dx = \frac{du}{3}$$$。

該積分可改寫為

$$84 i n t {\color{red}{\int{\sin{\left(3 x \right)} \cos{\left(3 x \right)} d x}}} = 84 i n t {\color{red}{\int{\frac{u}{3} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{3}$$$ 與 $$$f{\left(u \right)} = u$$$：

$$84 i n t {\color{red}{\int{\frac{u}{3} d u}}} = 84 i n t {\color{red}{\left(\frac{\int{u d u}}{3}\right)}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：

$$28 i n t {\color{red}{\int{u d u}}}=28 i n t {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}=28 i n t {\color{red}{\left(\frac{u^{2}}{2}\right)}}$$

回顧一下 $$$u=\sin{\left(3 x \right)}$$$：

$$14 i n t {\color{red}{u}}^{2} = 14 i n t {\color{red}{\sin{\left(3 x \right)}}}^{2}$$

因此，

$$\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x} = 14 i n t \sin^{2}{\left(3 x \right)}$$

加上積分常數：

$$\int{84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)} d x} = 14 i n t \sin^{2}{\left(3 x \right)}+C$$

$$$\int 84 i n t \sin{\left(3 x \right)} \cos{\left(3 x \right)}\, dx = 14 i n t \sin^{2}{\left(3 x \right)} + C$$$A