Integraal van $$$x \sin{\left(p \sqrt{- u^{2} + x^{2}} \right)}$$$ met betrekking tot $$$x$$$

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

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

Dan $$$dv=\left(- u^{2} + x^{2}\right)^{\prime }dx = 2 x dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$x dx = \frac{dv}{2}$$$.

De integraal kan worden herschreven als

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

Pas de constante-veelvoudregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(v \right)} = \sin{\left(p \sqrt{v} \right)}$$$:

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

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

Dan $$$dw=\left(p \sqrt{v}\right)^{\prime }dv = \frac{p}{2 \sqrt{v}} dv$$$ (de stappen zijn te zien »), en dan geldt dat $$$\frac{dv}{\sqrt{v}} = \frac{2 dw}{p}$$$.

De integraal kan worden herschreven als

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

Pas de constante-veelvoudregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ toe met $$$c=\frac{2}{p^{2}}$$$ en $$$f{\left(w \right)} = w \sin{\left(w \right)}$$$:

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

Voor de integraal $$$\int{w \sin{\left(w \right)} d w}$$$, gebruik partiële integratie $$$\int \operatorname{h} \operatorname{dm} = \operatorname{h}\operatorname{m} - \int \operatorname{m} \operatorname{dh}$$$.

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

Dan $$$\operatorname{dh}=\left(w\right)^{\prime }dw=1 dw$$$ (de stappen zijn te zien ») en $$$\operatorname{m}=\int{\sin{\left(w \right)} d w}=- \cos{\left(w \right)}$$$ (de stappen zijn te zien »).

De integraal wordt

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

Pas de constante-veelvoudregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ toe met $$$c=-1$$$ en $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

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

De integraal van de cosinus is $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

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

We herinneren eraan dat $$$w=p \sqrt{v}$$$:

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

We herinneren eraan dat $$$v=- u^{2} + x^{2}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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