Integraal van $$$\frac{\ln\left(x\right)}{\sqrt{x}}$$$

Bepaal $$$\int \frac{\ln\left(x\right)}{\sqrt{x}}\, dx$$$.

Voor de integraal $$$\int{\frac{\ln{\left(x \right)}}{\sqrt{x}} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=\ln{\left(x \right)}$$$ en $$$\operatorname{dv}=\frac{dx}{\sqrt{x}}$$$.

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{\frac{1}{\sqrt{x}} d x}=2 \sqrt{x}$$$ (de stappen zijn te zien »).

Dus,

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{\sqrt{x}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot 2 \sqrt{x}-\int{2 \sqrt{x} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(2 \sqrt{x} \ln{\left(x \right)} - \int{\frac{2}{\sqrt{x}} d x}\right)}}$$

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

$$2 \sqrt{x} \ln{\left(x \right)} - {\color{red}{\int{\frac{2}{\sqrt{x}} d x}}} = 2 \sqrt{x} \ln{\left(x \right)} - {\color{red}{\left(2 \int{\frac{1}{\sqrt{x}} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=- \frac{1}{2}$$$:

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

Dus,

$$\int{\frac{\ln{\left(x \right)}}{\sqrt{x}} d x} = 2 \sqrt{x} \ln{\left(x \right)} - 4 \sqrt{x}$$

Vereenvoudig:

$$\int{\frac{\ln{\left(x \right)}}{\sqrt{x}} d x} = 2 \sqrt{x} \left(\ln{\left(x \right)} - 2\right)$$

Voeg de integratieconstante toe:

$$\int{\frac{\ln{\left(x \right)}}{\sqrt{x}} d x} = 2 \sqrt{x} \left(\ln{\left(x \right)} - 2\right)+C$$

$$$\int \frac{\ln\left(x\right)}{\sqrt{x}}\, dx = 2 \sqrt{x} \left(\ln\left(x\right) - 2\right) + C$$$A