Integral de $$$\sqrt{\frac{x}{4 - x^{3}}}$$$

Encontre $$$\int \sqrt{\frac{x}{4 - x^{3}}}\, dx$$$.

A entrada é reescrita como: $$$\int{\sqrt{\frac{x}{4 - x^{3}}} d x}=\int{\frac{\sqrt{x}}{\sqrt{4 - x^{3}}} d x}$$$.

Seja $$$u=x^{\frac{3}{2}}$$$.

Então $$$du=\left(x^{\frac{3}{2}}\right)^{\prime }dx = \frac{3 \sqrt{x}}{2} dx$$$ (veja os passos »), e obtemos $$$\sqrt{x} dx = \frac{2 du}{3}$$$.

A integral pode ser reescrita como

$${\color{red}{\int{\frac{\sqrt{x}}{\sqrt{4 - x^{3}}} d x}}} = {\color{red}{\int{\frac{2}{3 \sqrt{4 - u^{2}}} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{2}{3}$$$ e $$$f{\left(u \right)} = \frac{1}{\sqrt{4 - u^{2}}}$$$:

$${\color{red}{\int{\frac{2}{3 \sqrt{4 - u^{2}}} d u}}} = {\color{red}{\left(\frac{2 \int{\frac{1}{\sqrt{4 - u^{2}}} d u}}{3}\right)}}$$

Seja $$$u=2 \sin{\left(v \right)}$$$.

Então $$$du=\left(2 \sin{\left(v \right)}\right)^{\prime }dv = 2 \cos{\left(v \right)} dv$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$v=\operatorname{asin}{\left(\frac{u}{2} \right)}$$$.

Portanto,

$$$\frac{1}{\sqrt{4 - u ^{2}}} = \frac{1}{\sqrt{4 - 4 \sin^{2}{\left( v \right)}}}$$$

Use a identidade $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\frac{1}{\sqrt{4 - 4 \sin^{2}{\left( v \right)}}}=\frac{1}{2 \sqrt{1 - \sin^{2}{\left( v \right)}}}=\frac{1}{2 \sqrt{\cos^{2}{\left( v \right)}}}$$$

Supondo que $$$\cos{\left( v \right)} \ge 0$$$, obtemos o seguinte:

$$$\frac{1}{2 \sqrt{\cos^{2}{\left( v \right)}}} = \frac{1}{2 \cos{\left( v \right)}}$$$

Portanto,

$$\frac{2 {\color{red}{\int{\frac{1}{\sqrt{4 - u^{2}}} d u}}}}{3} = \frac{2 {\color{red}{\int{1 d v}}}}{3}$$

Aplique a regra da constante $$$\int c\, dv = c v$$$ usando $$$c=1$$$:

$$\frac{2 {\color{red}{\int{1 d v}}}}{3} = \frac{2 {\color{red}{v}}}{3}$$

Recorde que $$$v=\operatorname{asin}{\left(\frac{u}{2} \right)}$$$:

$$\frac{2 {\color{red}{v}}}{3} = \frac{2 {\color{red}{\operatorname{asin}{\left(\frac{u}{2} \right)}}}}{3}$$

Recorde que $$$u=x^{\frac{3}{2}}$$$:

$$\frac{2 \operatorname{asin}{\left(\frac{{\color{red}{u}}}{2} \right)}}{3} = \frac{2 \operatorname{asin}{\left(\frac{{\color{red}{x^{\frac{3}{2}}}}}{2} \right)}}{3}$$

Portanto,

$$\int{\frac{\sqrt{x}}{\sqrt{4 - x^{3}}} d x} = \frac{2 \operatorname{asin}{\left(\frac{x^{\frac{3}{2}}}{2} \right)}}{3}$$

Adicione a constante de integração:

$$\int{\frac{\sqrt{x}}{\sqrt{4 - x^{3}}} d x} = \frac{2 \operatorname{asin}{\left(\frac{x^{\frac{3}{2}}}{2} \right)}}{3}+C$$

$$$\int \sqrt{\frac{x}{4 - x^{3}}}\, dx = \frac{2 \operatorname{asin}{\left(\frac{x^{\frac{3}{2}}}{2} \right)}}{3} + C$$$A