Integral de $$$\frac{5}{\sqrt{9 - 4 x^{2}}}$$$

Encontre $$$\int \frac{5}{\sqrt{9 - 4 x^{2}}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=5$$$ e $$$f{\left(x \right)} = \frac{1}{\sqrt{9 - 4 x^{2}}}$$$:

$${\color{red}{\int{\frac{5}{\sqrt{9 - 4 x^{2}}} d x}}} = {\color{red}{\left(5 \int{\frac{1}{\sqrt{9 - 4 x^{2}}} d x}\right)}}$$

Seja $$$x=\frac{3 \sin{\left(u \right)}}{2}$$$.

Então $$$dx=\left(\frac{3 \sin{\left(u \right)}}{2}\right)^{\prime }du = \frac{3 \cos{\left(u \right)}}{2} du$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$u=\operatorname{asin}{\left(\frac{2 x}{3} \right)}$$$.

Logo,

$$$\frac{1}{\sqrt{9 - 4 x^{2}}} = \frac{1}{\sqrt{9 - 9 \sin^{2}{\left( u \right)}}}$$$

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

$$$\frac{1}{\sqrt{9 - 9 \sin^{2}{\left( u \right)}}}=\frac{1}{3 \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{3 \sqrt{\cos^{2}{\left( u \right)}}}$$$

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

$$$\frac{1}{3 \sqrt{\cos^{2}{\left( u \right)}}} = \frac{1}{3 \cos{\left( u \right)}}$$$

Logo,

$$5 {\color{red}{\int{\frac{1}{\sqrt{9 - 4 x^{2}}} d x}}} = 5 {\color{red}{\int{\frac{1}{2} d u}}}$$

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=\frac{1}{2}$$$:

$$5 {\color{red}{\int{\frac{1}{2} d u}}} = 5 {\color{red}{\left(\frac{u}{2}\right)}}$$

Recorde que $$$u=\operatorname{asin}{\left(\frac{2 x}{3} \right)}$$$:

$$\frac{5 {\color{red}{u}}}{2} = \frac{5 {\color{red}{\operatorname{asin}{\left(\frac{2 x}{3} \right)}}}}{2}$$

Portanto,

$$\int{\frac{5}{\sqrt{9 - 4 x^{2}}} d x} = \frac{5 \operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2}$$

Adicione a constante de integração:

$$\int{\frac{5}{\sqrt{9 - 4 x^{2}}} d x} = \frac{5 \operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2}+C$$

$$$\int \frac{5}{\sqrt{9 - 4 x^{2}}}\, dx = \frac{5 \operatorname{asin}{\left(\frac{2 x}{3} \right)}}{2} + C$$$A