Integraal van $$$\sqrt{\frac{x}{4 - x^{3}}}$$$

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

De invoer is herschreven: $$$\int{\sqrt{\frac{x}{4 - x^{3}}} d x}=\int{\frac{\sqrt{x}}{\sqrt{4 - x^{3}}} d x}$$$.

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

Dan $$$du=\left(x^{\frac{3}{2}}\right)^{\prime }dx = \frac{3 \sqrt{x}}{2} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$\sqrt{x} dx = \frac{2 du}{3}$$$.

De integraal wordt

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{2}{3}$$$ en $$$f{\left(u \right)} = \frac{1}{\sqrt{4 - u^{2}}}$$$:

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

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

Dan $$$du=\left(2 \sin{\left(v \right)}\right)^{\prime }dv = 2 \cos{\left(v \right)} dv$$$ (zie » voor de stappen).

Bovendien volgt dat $$$v=\operatorname{asin}{\left(\frac{u}{2} \right)}$$$.

De integraand wordt

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

Gebruik de identiteit $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

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

Aangenomen dat $$$\cos{\left( v \right)} \ge 0$$$, verkrijgen we het volgende:

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

Dus,

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

Pas de constantenregel $$$\int c\, dv = c v$$$ toe met $$$c=1$$$:

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

We herinneren eraan dat $$$v=\operatorname{asin}{\left(\frac{u}{2} \right)}$$$:

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

We herinneren eraan dat $$$u=x^{\frac{3}{2}}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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