Funktion $$$7 x^{2} \sin{\left(x \right)}$$$ integraali

Määritä $$$\int 7 x^{2} \sin{\left(x \right)}\, dx$$$.

Sovella vakiokertoimen sääntöä $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ käyttäen $$$c=7$$$ ja $$$f{\left(x \right)} = x^{2} \sin{\left(x \right)}$$$:

$${\color{red}{\int{7 x^{2} \sin{\left(x \right)} d x}}} = {\color{red}{\left(7 \int{x^{2} \sin{\left(x \right)} d x}\right)}}$$

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

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

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

Näin ollen,

$$7 {\color{red}{\int{x^{2} \sin{\left(x \right)} d x}}}=7 {\color{red}{\left(x^{2} \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 2 x d x}\right)}}=7 {\color{red}{\left(- x^{2} \cos{\left(x \right)} - \int{\left(- 2 x \cos{\left(x \right)}\right)d x}\right)}}$$

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

$$- 7 x^{2} \cos{\left(x \right)} - 7 {\color{red}{\int{\left(- 2 x \cos{\left(x \right)}\right)d x}}} = - 7 x^{2} \cos{\left(x \right)} - 7 {\color{red}{\left(- 2 \int{x \cos{\left(x \right)} d x}\right)}}$$

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

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

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

Näin ollen,

$$- 7 x^{2} \cos{\left(x \right)} + 14 {\color{red}{\int{x \cos{\left(x \right)} d x}}}=- 7 x^{2} \cos{\left(x \right)} + 14 {\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}=- 7 x^{2} \cos{\left(x \right)} + 14 {\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 14 {\color{red}{\int{\sin{\left(x \right)} d x}}} = - 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} - 14 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Näin ollen,

$$\int{7 x^{2} \sin{\left(x \right)} d x} = - 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}$$

Lisää integrointivakio:

$$\int{7 x^{2} \sin{\left(x \right)} d x} = - 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}+C$$

$$$\int 7 x^{2} \sin{\left(x \right)}\, dx = \left(- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}\right) + C$$$A