$$$\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}$$$'nin integrali

Bulun: $$$\int \frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}\, dt$$$.

Sabit katsayı kuralı $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$'i $$$c=\frac{\sqrt{3}}{3}$$$ ve $$$f{\left(t \right)} = \frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}}$$$ ile uygula:

$${\color{red}{\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}} d t}}{3}\right)}}$$

$$$u=\sqrt{3} \sqrt{t}$$$ olsun.

Böylece $$$du=\left(\sqrt{3} \sqrt{t}\right)^{\prime }dt = \frac{\sqrt{3}}{2 \sqrt{t}} dt$$$ (adımlar » görülebilir) ve $$$\frac{dt}{\sqrt{t}} = \frac{2 \sqrt{3} du}{3}$$$ elde ederiz.

O halde,

$$\frac{\sqrt{3} {\color{red}{\int{\frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}} d t}}}}{3} = \frac{\sqrt{3} {\color{red}{\int{\frac{2 \sqrt{3} \cos{\left(u \right)}}{3} d u}}}}{3}$$

Sabit katsayı kuralı $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$'i $$$c=\frac{2 \sqrt{3}}{3}$$$ ve $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ ile uygula:

$$\frac{\sqrt{3} {\color{red}{\int{\frac{2 \sqrt{3} \cos{\left(u \right)}}{3} d u}}}}{3} = \frac{\sqrt{3} {\color{red}{\left(\frac{2 \sqrt{3} \int{\cos{\left(u \right)} d u}}{3}\right)}}}{3}$$

Kosinüsün integrali $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{2 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{3} = \frac{2 {\color{red}{\sin{\left(u \right)}}}}{3}$$

Hatırlayın ki $$$u=\sqrt{3} \sqrt{t}$$$:

$$\frac{2 \sin{\left({\color{red}{u}} \right)}}{3} = \frac{2 \sin{\left({\color{red}{\sqrt{3} \sqrt{t}}} \right)}}{3}$$

Dolayısıyla,

$$\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t} = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3}$$

İntegrasyon sabitini ekleyin:

$$\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t} = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3}+C$$

$$$\int \frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}\, dt = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3} + C$$$A