Integralen av $$$\frac{z \operatorname{asin}{\left(\ln\left(x\right) \right)}}{x}$$$ med avseende på $$$x$$$

Bestäm $$$\int \frac{z \operatorname{asin}{\left(\ln\left(x\right) \right)}}{x}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=z$$$ och $$$f{\left(x \right)} = \frac{\operatorname{asin}{\left(\ln{\left(x \right)} \right)}}{x}$$$:

$${\color{red}{\int{\frac{z \operatorname{asin}{\left(\ln{\left(x \right)} \right)}}{x} d x}}} = {\color{red}{z \int{\frac{\operatorname{asin}{\left(\ln{\left(x \right)} \right)}}{x} d x}}}$$

Låt $$$u=\ln{\left(x \right)}$$$ vara.

Då $$$du=\left(\ln{\left(x \right)}\right)^{\prime }dx = \frac{dx}{x}$$$ (stegen kan ses »), och vi har att $$$\frac{dx}{x} = du$$$.

Integralen kan omskrivas som

$$z {\color{red}{\int{\frac{\operatorname{asin}{\left(\ln{\left(x \right)} \right)}}{x} d x}}} = z {\color{red}{\int{\operatorname{asin}{\left(u \right)} d u}}}$$

För integralen $$$\int{\operatorname{asin}{\left(u \right)} d u}$$$, använd partiell integration $$$\int \operatorname{a} \operatorname{dv} = \operatorname{a}\operatorname{v} - \int \operatorname{v} \operatorname{da}$$$.

Låt $$$\operatorname{a}=\operatorname{asin}{\left(u \right)}$$$ och $$$\operatorname{dv}=du$$$.

Då gäller $$$\operatorname{da}=\left(\operatorname{asin}{\left(u \right)}\right)^{\prime }du=\frac{du}{\sqrt{1 - u^{2}}}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{1 d u}=u$$$ (stegen kan ses »).

Integralen kan omskrivas som

$$z {\color{red}{\int{\operatorname{asin}{\left(u \right)} d u}}}=z {\color{red}{\left(\operatorname{asin}{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{\sqrt{1 - u^{2}}} d u}\right)}}=z {\color{red}{\left(u \operatorname{asin}{\left(u \right)} - \int{\frac{u}{\sqrt{1 - u^{2}}} d u}\right)}}$$

Låt $$$v=1 - u^{2}$$$ vara.

Då $$$dv=\left(1 - u^{2}\right)^{\prime }du = - 2 u du$$$ (stegen kan ses »), och vi har att $$$u du = - \frac{dv}{2}$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=- \frac{1}{2}$$$ och $$$f{\left(v \right)} = \frac{1}{\sqrt{v}}$$$:

$$z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\int{\left(- \frac{1}{2 \sqrt{v}}\right)d v}}}\right) = z \left(u \operatorname{asin}{\left(u \right)} - {\color{red}{\left(- \frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}\right)$$

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=- \frac{1}{2}$$$:

$$z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{2}\right)=z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{2}\right)=z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}\right)=z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{2}\right)=z \left(u \operatorname{asin}{\left(u \right)} + \frac{{\color{red}{\left(2 \sqrt{v}\right)}}}{2}\right)$$

Kom ihåg att $$$v=1 - u^{2}$$$:

$$z \left(u \operatorname{asin}{\left(u \right)} + \sqrt{{\color{red}{v}}}\right) = z \left(u \operatorname{asin}{\left(u \right)} + \sqrt{{\color{red}{\left(1 - u^{2}\right)}}}\right)$$

Kom ihåg att $$$u=\ln{\left(x \right)}$$$:

$$z \left(\sqrt{1 - {\color{red}{u}}^{2}} + {\color{red}{u}} \operatorname{asin}{\left({\color{red}{u}} \right)}\right) = z \left(\sqrt{1 - {\color{red}{\ln{\left(x \right)}}}^{2}} + {\color{red}{\ln{\left(x \right)}}} \operatorname{asin}{\left({\color{red}{\ln{\left(x \right)}}} \right)}\right)$$

Alltså,

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

Lägg till integrationskonstanten:

$$\int{\frac{z \operatorname{asin}{\left(\ln{\left(x \right)} \right)}}{x} d x} = z \left(\sqrt{1 - \ln{\left(x \right)}^{2}} + \ln{\left(x \right)} \operatorname{asin}{\left(\ln{\left(x \right)} \right)}\right)+C$$

$$$\int \frac{z \operatorname{asin}{\left(\ln\left(x\right) \right)}}{x}\, dx = z \left(\sqrt{1 - \ln^{2}\left(x\right)} + \ln\left(x\right) \operatorname{asin}{\left(\ln\left(x\right) \right)}\right) + C$$$A