$$$\sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}$$$ 的積分

求$$$\int \sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}\, dx$$$。

先提出一個餘弦，並將其餘部分以正弦表示，使用公式 $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$，令 $$$\alpha=x$$$:

$${\color{red}{\int{\sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} d x}}} = {\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right) \sqrt[4]{\sin{\left(x \right)}} \cos{\left(x \right)} d x}}}$$

令 $$$u=\sin{\left(x \right)}$$$。

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

該積分變為

$${\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right) \sqrt[4]{\sin{\left(x \right)}} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\sqrt[4]{u} \left(1 - u^{2}\right) d u}}}$$

Expand the expression:

$${\color{red}{\int{\sqrt[4]{u} \left(1 - u^{2}\right) d u}}} = {\color{red}{\int{\left(- u^{\frac{9}{4}} + \sqrt[4]{u}\right)d u}}}$$

逐項積分：

$${\color{red}{\int{\left(- u^{\frac{9}{4}} + \sqrt[4]{u}\right)d u}}} = {\color{red}{\left(\int{\sqrt[4]{u} d u} - \int{u^{\frac{9}{4}} d u}\right)}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=\frac{1}{4}$$$：

$$- \int{u^{\frac{9}{4}} d u} + {\color{red}{\int{\sqrt[4]{u} d u}}}=- \int{u^{\frac{9}{4}} d u} + {\color{red}{\int{u^{\frac{1}{4}} d u}}}=- \int{u^{\frac{9}{4}} d u} + {\color{red}{\frac{u^{\frac{1}{4} + 1}}{\frac{1}{4} + 1}}}=- \int{u^{\frac{9}{4}} d u} + {\color{red}{\left(\frac{4 u^{\frac{5}{4}}}{5}\right)}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=\frac{9}{4}$$$：

$$\frac{4 u^{\frac{5}{4}}}{5} - {\color{red}{\int{u^{\frac{9}{4}} d u}}}=\frac{4 u^{\frac{5}{4}}}{5} - {\color{red}{\frac{u^{1 + \frac{9}{4}}}{1 + \frac{9}{4}}}}=\frac{4 u^{\frac{5}{4}}}{5} - {\color{red}{\left(\frac{4 u^{\frac{13}{4}}}{13}\right)}}$$

回顧一下 $$$u=\sin{\left(x \right)}$$$：

$$\frac{4 {\color{red}{u}}^{\frac{5}{4}}}{5} - \frac{4 {\color{red}{u}}^{\frac{13}{4}}}{13} = \frac{4 {\color{red}{\sin{\left(x \right)}}}^{\frac{5}{4}}}{5} - \frac{4 {\color{red}{\sin{\left(x \right)}}}^{\frac{13}{4}}}{13}$$

因此，

$$\int{\sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} d x} = - \frac{4 \sin^{\frac{13}{4}}{\left(x \right)}}{13} + \frac{4 \sin^{\frac{5}{4}}{\left(x \right)}}{5}$$

化簡：

$$\int{\sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} d x} = \frac{4 \left(13 - 5 \sin^{2}{\left(x \right)}\right) \sin^{\frac{5}{4}}{\left(x \right)}}{65}$$

加上積分常數：

$$\int{\sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)} d x} = \frac{4 \left(13 - 5 \sin^{2}{\left(x \right)}\right) \sin^{\frac{5}{4}}{\left(x \right)}}{65}+C$$

$$$\int \sqrt[4]{\sin{\left(x \right)}} \cos^{3}{\left(x \right)}\, dx = \frac{4 \left(13 - 5 \sin^{2}{\left(x \right)}\right) \sin^{\frac{5}{4}}{\left(x \right)}}{65} + C$$$A