Integralen av $$$x^{2} \sin{\left(x^{3} \right)}$$$

Bestäm $$$\int x^{2} \sin{\left(x^{3} \right)}\, dx$$$.

Låt $$$u=x^{3}$$$ vara.

Då $$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (stegen kan ses »), och vi har att $$$x^{2} dx = \frac{du}{3}$$$.

Integralen blir

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{3}\right)}}$$

Integralen av sinus är $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{3} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{3}$$

Kom ihåg att $$$u=x^{3}$$$:

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

Alltså,

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

Lägg till integrationskonstanten:

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

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