Integral de $$$2 x \cos{\left(x^{2} \right)}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int 2 x \cos{\left(x^{2} \right)}\, dx$$$.
Solución
Sea $$$u=x^{2}$$$.
Entonces $$$du=\left(x^{2}\right)^{\prime }dx = 2 x dx$$$ (los pasos pueden verse »), y obtenemos que $$$x dx = \frac{du}{2}$$$.
La integral se convierte en
$${\color{red}{\int{2 x \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\int{\cos{\left(u \right)} d u}}}$$
La integral del coseno es $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:
$${\color{red}{\int{\cos{\left(u \right)} d u}}} = {\color{red}{\sin{\left(u \right)}}}$$
Recordemos que $$$u=x^{2}$$$:
$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{x^{2}}} \right)}$$
Por lo tanto,
$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}$$
Añade la constante de integración:
$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}+C$$
Respuesta
$$$\int 2 x \cos{\left(x^{2} \right)}\, dx = \sin{\left(x^{2} \right)} + C$$$A