Integral de $$$\cos{\left(2 x y \right)}$$$ con respecto a $$$x$$$

Halla $$$\int \cos{\left(2 x y \right)}\, dx$$$.

Sea $$$u=2 x y$$$.

Entonces $$$du=\left(2 x y\right)^{\prime }dx = 2 y dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2 y}$$$.

La integral se convierte en

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2 y}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

La integral del coseno es $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

Recordemos que $$$u=2 x y$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{2 y} = \frac{\sin{\left({\color{red}{\left(2 x y\right)}} \right)}}{2 y}$$

Por lo tanto,

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

Añade la constante de integración:

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

$$$\int \cos{\left(2 x y \right)}\, dx = \frac{\sin{\left(2 x y \right)}}{2 y} + C$$$A