$$$\tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)}$$$ 的積分

求$$$\int \tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)}\, dy$$$。

提出兩個正割，並使用公式 $$$\sec^2\left( \alpha \right)=\tan^2\left( \alpha \right) + 1$$$（其中 $$$\alpha=y$$$），將其餘全部用正切表示:

$${\color{red}{\int{\tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)} d y}}} = {\color{red}{\int{\left(\tan^{2}{\left(y \right)} + 1\right) \tan^{4}{\left(y \right)} \sec^{2}{\left(y \right)} d y}}}$$

令 $$$u=\tan{\left(y \right)}$$$。

則 $$$du=\left(\tan{\left(y \right)}\right)^{\prime }dy = \sec^{2}{\left(y \right)} dy$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(y \right)} dy = du$$$。

因此，

$${\color{red}{\int{\left(\tan^{2}{\left(y \right)} + 1\right) \tan^{4}{\left(y \right)} \sec^{2}{\left(y \right)} d y}}} = {\color{red}{\int{u^{4} \left(u^{2} + 1\right) d u}}}$$

Expand the expression:

$${\color{red}{\int{u^{4} \left(u^{2} + 1\right) d u}}} = {\color{red}{\int{\left(u^{6} + u^{4}\right)d u}}}$$

逐項積分：

$${\color{red}{\int{\left(u^{6} + u^{4}\right)d u}}} = {\color{red}{\left(\int{u^{4} d u} + \int{u^{6} d u}\right)}}$$

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

$$\int{u^{6} d u} + {\color{red}{\int{u^{4} d u}}}=\int{u^{6} d u} + {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=\int{u^{6} d u} + {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=6$$$：

$$\frac{u^{5}}{5} + {\color{red}{\int{u^{6} d u}}}=\frac{u^{5}}{5} + {\color{red}{\frac{u^{1 + 6}}{1 + 6}}}=\frac{u^{5}}{5} + {\color{red}{\left(\frac{u^{7}}{7}\right)}}$$

回顧一下 $$$u=\tan{\left(y \right)}$$$：

$$\frac{{\color{red}{u}}^{5}}{5} + \frac{{\color{red}{u}}^{7}}{7} = \frac{{\color{red}{\tan{\left(y \right)}}}^{5}}{5} + \frac{{\color{red}{\tan{\left(y \right)}}}^{7}}{7}$$

因此，

$$\int{\tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)} d y} = \frac{\tan^{7}{\left(y \right)}}{7} + \frac{\tan^{5}{\left(y \right)}}{5}$$

加上積分常數：

$$\int{\tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)} d y} = \frac{\tan^{7}{\left(y \right)}}{7} + \frac{\tan^{5}{\left(y \right)}}{5}+C$$

$$$\int \tan^{4}{\left(y \right)} \sec^{4}{\left(y \right)}\, dy = \left(\frac{\tan^{7}{\left(y \right)}}{7} + \frac{\tan^{5}{\left(y \right)}}{5}\right) + C$$$A