Integral dari $$$2 \cos{\left(x^{2} \right)}$$$

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=2$$$ dan $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:

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

Integral ini (Integral Kosinus Fresnel) tidak memiliki bentuk tertutup:

$$2 {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} = 2 {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$

Oleh karena itu,

$$\int{2 \cos{\left(x^{2} \right)} d x} = \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)$$

Tambahkan konstanta integrasi:

$$\int{2 \cos{\left(x^{2} \right)} d x} = \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)+C$$

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