Integral de $$$i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1$$$ em relação a $$$x$$$

Encontre $$$\int \left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)\, dx$$$.

A entrada é reescrita como: $$$\int{\left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)d x}=\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x}$$$.

Integre termo a termo:

$${\color{red}{\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

$$- \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{\int{1 d x}}} = - \int{y^{2} d x} - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{x}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=y^{2}$$$:

$$- x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{\int{y^{2} d x}}} = - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + \int{i n t d x} - {\color{red}{x y^{2}}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=i n t$$$:

$$- x y^{2} - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + {\color{red}{\int{i n t d x}}} = - x y^{2} - x - \int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x} + {\color{red}{i n t x}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\sqrt{2} y^{2}$$$ e $$$f{\left(x \right)} = x^{\frac{7}{2}}$$$:

$$i n t x - x y^{2} - x - {\color{red}{\int{\sqrt{2} x^{\frac{7}{2}} y^{2} d x}}} = i n t x - x y^{2} - x - {\color{red}{\sqrt{2} y^{2} \int{x^{\frac{7}{2}} d x}}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{7}{2}$$$:

$$i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\int{x^{\frac{7}{2}} d x}}}=i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\frac{x^{1 + \frac{7}{2}}}{1 + \frac{7}{2}}}}=i n t x - x y^{2} - x - \sqrt{2} y^{2} {\color{red}{\left(\frac{2 x^{\frac{9}{2}}}{9}\right)}}$$

Portanto,

$$\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x} = i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x$$

Adicione a constante de integração:

$$\int{\left(i n t - \sqrt{2} x^{\frac{7}{2}} y^{2} - y^{2} - 1\right)d x} = i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x+C$$

$$$\int \left(i n t - \sqrt{2} y^{2} \sqrt{x^{7}} - y^{2} - 1\right)\, dx = \left(i n t x - \frac{2 \sqrt{2} x^{\frac{9}{2}} y^{2}}{9} - x y^{2} - x\right) + C$$$A