Integralen av $$$\operatorname{asin}{\left(x \right)}$$$

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

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

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

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

Alltså,

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

Låt $$$u=1 - x^{2}$$$ vara.

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

Alltså,

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

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

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

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

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

Kom ihåg att $$$u=1 - x^{2}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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