Funktion $$$\frac{x^{2} \cos{\left(x \right)}}{2}$$$ integraali

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=\frac{1}{2}$$$ ja $$$f{\left(x \right)} = x^{2} \cos{\left(x \right)}$$$:

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

Integraalin $$$\int{x^{2} \cos{\left(x \right)} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Olkoon $$$\operatorname{u}=x^{2}$$$ ja $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$.

Tällöin $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$ (vaiheet ovat nähtävissä »).

Näin ollen,

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=2$$$ ja $$$f{\left(x \right)} = x \sin{\left(x \right)}$$$:

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

Integraalin $$$\int{x \sin{\left(x \right)} d x}$$$ kohdalla käytä osittaisintegrointia $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Olkoon $$$\operatorname{u}=x$$$ ja $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$.

Tällöin $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (vaiheet ovat nähtävissä ») ja $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (vaiheet ovat nähtävissä »).

Integraali muuttuu muotoon

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

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=-1$$$ ja $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

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

Kosinin integraali on $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Näin ollen,

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

Lisää integrointivakio:

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

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