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

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

För integralen $$$\int{x^{2} \cos{\left(3 x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=x^{2}$$$ och $$$\operatorname{dv}=\cos{\left(3 x \right)} dx$$$.

Då gäller $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\cos{\left(3 x \right)} d x}=\frac{\sin{\left(3 x \right)}}{3}$$$ (stegen kan ses »).

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{2}{3}$$$ och $$$f{\left(x \right)} = x \sin{\left(3 x \right)}$$$:

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

För integralen $$$\int{x \sin{\left(3 x \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=x$$$ och $$$\operatorname{dv}=\sin{\left(3 x \right)} dx$$$.

Då gäller $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{\sin{\left(3 x \right)} d x}=- \frac{\cos{\left(3 x \right)}}{3}$$$ (stegen kan ses »).

Alltså,

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\int{x \sin{\left(3 x \right)} d x}}}}{3}=\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\left(x \cdot \left(- \frac{\cos{\left(3 x \right)}}{3}\right)-\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right) \cdot 1 d x}\right)}}}{3}=\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\left(- \frac{x \cos{\left(3 x \right)}}{3} - \int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}\right)}}}{3}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{1}{3}$$$ och $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$:

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} + \frac{2 {\color{red}{\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}}}}{3} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} + \frac{2 {\color{red}{\left(- \frac{\int{\cos{\left(3 x \right)} d x}}{3}\right)}}}{3}$$

Låt $$$u=3 x$$$ vara.

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

Alltså,

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{9} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{9}$$

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

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{9} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{3}\right)}}}{9}$$

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

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{27} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\sin{\left(u \right)}}}}{27}$$

Kom ihåg att $$$u=3 x$$$:

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left({\color{red}{u}} \right)}}{27} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left({\color{red}{\left(3 x\right)}} \right)}}{27}$$

Alltså,

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left(3 x \right)}}{27}$$

Förenkla:

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27}$$

Lägg till integrationskonstanten:

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27}+C$$

$$$\int x^{2} \cos{\left(3 x \right)}\, dx = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27} + C$$$A