Integral dari $$$\frac{1}{\cos^{2}{\left(x y \right)}}$$$ terhadap $$$x$$$

Temukan $$$\int \frac{1}{\cos^{2}{\left(x y \right)}}\, dx$$$.

Misalkan $$$u=x y$$$.

Kemudian $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = \frac{du}{y}$$$.

Dengan demikian,

$${\color{red}{\int{\frac{1}{\cos^{2}{\left(x y \right)}} d x}}} = {\color{red}{\int{\frac{1}{y \cos^{2}{\left(u \right)}} d u}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{y}$$$ dan $$$f{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}$$$:

$${\color{red}{\int{\frac{1}{y \cos^{2}{\left(u \right)}} d u}}} = {\color{red}{\frac{\int{\frac{1}{\cos^{2}{\left(u \right)}} d u}}{y}}}$$

Tulis ulang integran dalam bentuk fungsi sekan.:

$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(u \right)}} d u}}}}{y} = \frac{{\color{red}{\int{\sec^{2}{\left(u \right)} d u}}}}{y}$$

Integral dari $$$\sec^{2}{\left(u \right)}$$$ adalah $$$\int{\sec^{2}{\left(u \right)} d u} = \tan{\left(u \right)}$$$:

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

Ingat bahwa $$$u=x y$$$:

$$\frac{\tan{\left({\color{red}{u}} \right)}}{y} = \frac{\tan{\left({\color{red}{x y}} \right)}}{y}$$

Oleh karena itu,

$$\int{\frac{1}{\cos^{2}{\left(x y \right)}} d x} = \frac{\tan{\left(x y \right)}}{y}$$

Tambahkan konstanta integrasi:

$$\int{\frac{1}{\cos^{2}{\left(x y \right)}} d x} = \frac{\tan{\left(x y \right)}}{y}+C$$

$$$\int \frac{1}{\cos^{2}{\left(x y \right)}}\, dx = \frac{\tan{\left(x y \right)}}{y} + C$$$A