Integral de $$$\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}\, dt$$$.
Solución
Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=\frac{\sqrt{3}}{3}$$$ y $$$f{\left(t \right)} = \frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}}$$$:
$${\color{red}{\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}} d t}}{3}\right)}}$$
Sea $$$u=\sqrt{3} \sqrt{t}$$$.
Entonces $$$du=\left(\sqrt{3} \sqrt{t}\right)^{\prime }dt = \frac{\sqrt{3}}{2 \sqrt{t}} dt$$$ (los pasos pueden verse »), y obtenemos que $$$\frac{dt}{\sqrt{t}} = \frac{2 \sqrt{3} du}{3}$$$.
La integral puede reescribirse como
$$\frac{\sqrt{3} {\color{red}{\int{\frac{\cos{\left(\sqrt{3} \sqrt{t} \right)}}{\sqrt{t}} d t}}}}{3} = \frac{\sqrt{3} {\color{red}{\int{\frac{2 \sqrt{3} \cos{\left(u \right)}}{3} d u}}}}{3}$$
Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{2 \sqrt{3}}{3}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:
$$\frac{\sqrt{3} {\color{red}{\int{\frac{2 \sqrt{3} \cos{\left(u \right)}}{3} d u}}}}{3} = \frac{\sqrt{3} {\color{red}{\left(\frac{2 \sqrt{3} \int{\cos{\left(u \right)} d u}}{3}\right)}}}{3}$$
La integral del coseno es $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:
$$\frac{2 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{3} = \frac{2 {\color{red}{\sin{\left(u \right)}}}}{3}$$
Recordemos que $$$u=\sqrt{3} \sqrt{t}$$$:
$$\frac{2 \sin{\left({\color{red}{u}} \right)}}{3} = \frac{2 \sin{\left({\color{red}{\sqrt{3} \sqrt{t}}} \right)}}{3}$$
Por lo tanto,
$$\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t} = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3}$$
Añade la constante de integración:
$$\int{\frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}} d t} = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3}+C$$
Respuesta
$$$\int \frac{\sqrt{3} \cos{\left(\sqrt{3} \sqrt{t} \right)}}{3 \sqrt{t}}\, dt = \frac{2 \sin{\left(\sqrt{3} \sqrt{t} \right)}}{3} + C$$$A