Integral de $$$x \sin{\left(p \sqrt{- u^{2} + x^{2}} \right)}$$$ em relação a $$$x$$$

Encontre $$$\int x \sin{\left(p \sqrt{- u^{2} + x^{2}} \right)}\, dx$$$.

Seja $$$v=- u^{2} + x^{2}$$$.

Então $$$dv=\left(- u^{2} + x^{2}\right)^{\prime }dx = 2 x dx$$$ (veja os passos »), e obtemos $$$x dx = \frac{dv}{2}$$$.

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(v \right)} = \sin{\left(p \sqrt{v} \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(p \sqrt{v} \right)}}{2} d v}}} = {\color{red}{\left(\frac{\int{\sin{\left(p \sqrt{v} \right)} d v}}{2}\right)}}$$

Seja $$$w=p \sqrt{v}$$$.

Então $$$dw=\left(p \sqrt{v}\right)^{\prime }dv = \frac{p}{2 \sqrt{v}} dv$$$ (veja os passos »), e obtemos $$$\frac{dv}{\sqrt{v}} = \frac{2 dw}{p}$$$.

Portanto,

$$\frac{{\color{red}{\int{\sin{\left(p \sqrt{v} \right)} d v}}}}{2} = \frac{{\color{red}{\int{\frac{2 w \sin{\left(w \right)}}{p^{2}} d w}}}}{2}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ usando $$$c=\frac{2}{p^{2}}$$$ e $$$f{\left(w \right)} = w \sin{\left(w \right)}$$$:

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

Para a integral $$$\int{w \sin{\left(w \right)} d w}$$$, use integração por partes $$$\int \operatorname{h} \operatorname{dm} = \operatorname{h}\operatorname{m} - \int \operatorname{m} \operatorname{dh}$$$.

Sejam $$$\operatorname{h}=w$$$ e $$$\operatorname{dm}=\sin{\left(w \right)} dw$$$.

Então $$$\operatorname{dh}=\left(w\right)^{\prime }dw=1 dw$$$ (os passos podem ser vistos ») e $$$\operatorname{m}=\int{\sin{\left(w \right)} d w}=- \cos{\left(w \right)}$$$ (os passos podem ser vistos »).

Assim,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ usando $$$c=-1$$$ e $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

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

A integral do cosseno é $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

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

Recorde que $$$w=p \sqrt{v}$$$:

$$\frac{\sin{\left({\color{red}{w}} \right)} - {\color{red}{w}} \cos{\left({\color{red}{w}} \right)}}{p^{2}} = \frac{\sin{\left({\color{red}{p \sqrt{v}}} \right)} - {\color{red}{p \sqrt{v}}} \cos{\left({\color{red}{p \sqrt{v}}} \right)}}{p^{2}}$$

Recorde que $$$v=- u^{2} + x^{2}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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