Integral de $$$x^{2} \sin{\left(n x \right)}$$$ em relação a $$$x$$$

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

Para a integral $$$\int{x^{2} \sin{\left(n x \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x^{2}$$$ e $$$\operatorname{dv}=\sin{\left(n x \right)} dx$$$.

Então $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\sin{\left(n x \right)} d x}=- \frac{\cos{\left(n x \right)}}{n}$$$ (os passos podem ser vistos »).

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=- \frac{2}{n}$$$ e $$$f{\left(x \right)} = x \cos{\left(n x \right)}$$$:

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

Para a integral $$$\int{x \cos{\left(n x \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x$$$ e $$$\operatorname{dv}=\cos{\left(n x \right)} dx$$$.

Então $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\cos{\left(n x \right)} d x}=\frac{\sin{\left(n x \right)}}{n}$$$ (os passos podem ser vistos »).

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{n}$$$ e $$$f{\left(x \right)} = \sin{\left(n x \right)}$$$:

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

Seja $$$u=n x$$$.

Então $$$du=\left(n x\right)^{\prime }dx = n dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{n}$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{n}$$$ e $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

A integral do seno é $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Recorde que $$$u=n x$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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