Integraal van $$$\sqrt{\frac{1 - t}{t}}$$$

Bepaal $$$\int \sqrt{\frac{1 - t}{t}}\, dt$$$.

De invoer is herschreven: $$$\int{\sqrt{\frac{1 - t}{t}} d t}=\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t}$$$.

Zij $$$u=\sqrt{t}$$$.

Dan $$$du=\left(\sqrt{t}\right)^{\prime }dt = \frac{1}{2 \sqrt{t}} dt$$$ (de stappen zijn te zien »), en dan geldt dat $$$\frac{dt}{\sqrt{t}} = 2 du$$$.

De integraal wordt

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

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

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

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

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

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

Dus,

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

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

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

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

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

De integraal kan worden herschreven als

$$2 {\color{red}{\int{\sqrt{1 - u^{2}} d u}}} = 2 {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}}$$

Pas de machtsreductieformule $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ toe met $$$\alpha= v $$$:

$$2 {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}} = 2 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(v \right)} = \cos{\left(2 v \right)} + 1$$$:

$$2 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}} = 2 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}{2}\right)}}$$

Integreer termgewijs:

$${\color{red}{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}} = {\color{red}{\left(\int{1 d v} + \int{\cos{\left(2 v \right)} d v}\right)}}$$

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

$$\int{\cos{\left(2 v \right)} d v} + {\color{red}{\int{1 d v}}} = \int{\cos{\left(2 v \right)} d v} + {\color{red}{v}}$$

Zij $$$w=2 v$$$.

Dan $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (de stappen zijn te zien »), en dan geldt dat $$$dv = \frac{dw}{2}$$$.

De integraal wordt

$$v + {\color{red}{\int{\cos{\left(2 v \right)} d v}}} = v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

$$v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}} = v + {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}$$

De integraal van de cosinus is $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

$$v + \frac{{\color{red}{\int{\cos{\left(w \right)} d w}}}}{2} = v + \frac{{\color{red}{\sin{\left(w \right)}}}}{2}$$

We herinneren eraan dat $$$w=2 v$$$:

$$v + \frac{\sin{\left({\color{red}{w}} \right)}}{2} = v + \frac{\sin{\left({\color{red}{\left(2 v\right)}} \right)}}{2}$$

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

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

We herinneren eraan dat $$$u=\sqrt{t}$$$:

$$\frac{\sin{\left(2 \operatorname{asin}{\left({\color{red}{u}} \right)} \right)}}{2} + \operatorname{asin}{\left({\color{red}{u}} \right)} = \frac{\sin{\left(2 \operatorname{asin}{\left({\color{red}{\sqrt{t}}} \right)} \right)}}{2} + \operatorname{asin}{\left({\color{red}{\sqrt{t}}} \right)}$$

Dus,

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \frac{\sin{\left(2 \operatorname{asin}{\left(\sqrt{t} \right)} \right)}}{2} + \operatorname{asin}{\left(\sqrt{t} \right)}$$

Gebruik de formules $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$ om de uitdrukking te vereenvoudigen:

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}+C$$

$$$\int \sqrt{\frac{1 - t}{t}}\, dt = \left(\sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}\right) + C$$$A