Integral de $$$\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)}$$$

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

Seja $$$u=\cos{\left(x \right)}$$$.

Então $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (veja os passos »), e obtemos $$$\sin{\left(x \right)} dx = - du$$$.

Logo,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-1$$$ e $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$:

$${\color{red}{\int{\left(- \sec^{2}{\left(u \right)}\right)d u}}} = {\color{red}{\left(- \int{\sec^{2}{\left(u \right)} d u}\right)}}$$

A integral de $$$\sec^{2}{\left(u \right)}$$$ é $$$\int{\sec^{2}{\left(u \right)} d u} = \tan{\left(u \right)}$$$:

$$- {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}} = - {\color{red}{\tan{\left(u \right)}}}$$

Recorde que $$$u=\cos{\left(x \right)}$$$:

$$- \tan{\left({\color{red}{u}} \right)} = - \tan{\left({\color{red}{\cos{\left(x \right)}}} \right)}$$

Portanto,

$$\int{\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)} d x} = - \tan{\left(\cos{\left(x \right)} \right)}$$

Adicione a constante de integração:

$$\int{\sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)} d x} = - \tan{\left(\cos{\left(x \right)} \right)}+C$$

$$$\int \sin{\left(x \right)} \sec^{2}{\left(\cos{\left(x \right)} \right)}\, dx = - \tan{\left(\cos{\left(x \right)} \right)} + C$$$A