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

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

Sederhanakan integran:

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

Untuk integral $$$\int{x \cos{\left(x \right)} d x}$$$, gunakan integrasi parsial $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Misalkan $$$\operatorname{u}=x$$$ dan $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$.

Maka $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (langkah-langkah dapat dilihat di ») dan $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$ (langkah-langkah dapat dilihat di »).

Oleh karena itu,

$${\color{red}{\int{x \cos{\left(x \right)} d x}}}={\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}={\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$

Integral dari sinus adalah $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$x \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = x \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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