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$$$e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)}$$$ 的積分
相關計算器: 定積分與廣義積分計算器
您的輸入
求$$$\int e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)}\, dx$$$。
解答
令 $$$u=\tan{\left(x \right)}$$$。
則 $$$du=\left(\tan{\left(x \right)}\right)^{\prime }dx = \sec^{2}{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(x \right)} dx = du$$$。
因此，
$${\color{red}{\int{e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)} d x}}} = {\color{red}{\int{e^{u} d u}}}$$
指數函數的積分為 $$$\int{e^{u} d u} = e^{u}$$$：
$${\color{red}{\int{e^{u} d u}}} = {\color{red}{e^{u}}}$$
回顧一下 $$$u=\tan{\left(x \right)}$$$：
$$e^{{\color{red}{u}}} = e^{{\color{red}{\tan{\left(x \right)}}}}$$
因此，
$$\int{e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)} d x} = e^{\tan{\left(x \right)}}$$
加上積分常數：
$$\int{e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)} d x} = e^{\tan{\left(x \right)}}+C$$
答案
$$$\int e^{\tan{\left(x \right)}} \sec^{2}{\left(x \right)}\, dx = e^{\tan{\left(x \right)}} + C$$$A