Integral dari $$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$$

Temukan $$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$.

Tulis ulang integran:

$${\color{red}{\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$

Misalkan $$$u=\sec{\left(x \right)}$$$.

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

Integral tersebut dapat ditulis ulang sebagai

$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$

Terapkan aturan konstanta $$$\int c\, du = c u$$$ dengan $$$c=1$$$:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Ingat bahwa $$$u=\sec{\left(x \right)}$$$:

$${\color{red}{u}} = {\color{red}{\sec{\left(x \right)}}}$$

Oleh karena itu,

$$\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x} = \sec{\left(x \right)}$$

Tambahkan konstanta integrasi:

$$\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x} = \sec{\left(x \right)}+C$$

$$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx = \sec{\left(x \right)} + C$$$A