Integralen av $$$\sin^{2}{\left(x \right)} \cos{\left(x \right)}$$$

Bestäm $$$\int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$$.

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 $$$\cos{\left(x \right)} dx = du$$$.

Alltså,

$${\color{red}{\int{\sin^{2}{\left(x \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\int{u^{2} d u}}}$$

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

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

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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