|
|
|
|
|
|
|
|
|
|
|
|
$$$1 - \sec^{2}{\left(x \right)}$$$ 的積分
您的輸入
求$$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\sec^{2}{\left(x \right)} d x}\right)}}$$
配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$$- \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\sec^{2}{\left(x \right)} d x} + {\color{red}{x}}$$
$$$\sec^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$：
$$x - {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = x - {\color{red}{\tan{\left(x \right)}}}$$
因此，
$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}$$
加上積分常數：
$$\int{\left(1 - \sec^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)}+C$$
答案
$$$\int \left(1 - \sec^{2}{\left(x \right)}\right)\, dx = \left(x - \tan{\left(x \right)}\right) + C$$$A