Integralen av $$$\sqrt{\sin{\left(x \right)}} \cos{\left(x \right)}$$$

Bestäm $$$\int \sqrt{\sin{\left(x \right)}} \cos{\left(x \right)}\, dx$$$.

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

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

Integralen kan omskrivas som

$${\color{red}{\int{\sqrt{\sin{\left(x \right)}} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\sqrt{u} d u}}}$$

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

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

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

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

Alltså,

$$\int{\sqrt{\sin{\left(x \right)}} \cos{\left(x \right)} d x} = \frac{2 \sin^{\frac{3}{2}}{\left(x \right)}}{3}$$

Lägg till integrationskonstanten:

$$\int{\sqrt{\sin{\left(x \right)}} \cos{\left(x \right)} d x} = \frac{2 \sin^{\frac{3}{2}}{\left(x \right)}}{3}+C$$

$$$\int \sqrt{\sin{\left(x \right)}} \cos{\left(x \right)}\, dx = \frac{2 \sin^{\frac{3}{2}}{\left(x \right)}}{3} + C$$$A