Integral dari $$$- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}$$$

Temukan $$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)\, dx$$$.

Integralkan suku demi suku:

$${\color{red}{\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{2} d x} - \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=\frac{1}{2}$$$:

$$- \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x} + {\color{red}{\int{\frac{1}{2} d x}}} = - \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x} + {\color{red}{\left(\frac{x}{2}\right)}}$$

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

$$\frac{x}{2} + \int{\cos{\left(x \right)} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \frac{x}{2} + \int{\cos{\left(x \right)} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

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

$$\frac{x}{2} + \cos{\left(x \right)} + {\color{red}{\int{\cos{\left(x \right)} d x}}} = \frac{x}{2} + \cos{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}$$

Oleh karena itu,

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sin{\left(x \right)} + \cos{\left(x \right)}$$

Sederhanakan:

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$

Tambahkan konstanta integrasi:

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$

$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)\, dx = \left(\frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}\right) + C$$$A