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

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

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

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

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

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

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

Näin ollen,

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

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

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

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

Näin ollen,

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

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

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

Olemme päätyneet integraaliin, jonka olemme jo aiemmin nähneet.

Näin ollen olemme saaneet seuraavan yksinkertaisen integraalia koskevan yhtälön:

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

Ratkaisemalla sen saamme, että

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

Näin ollen,

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

Näin ollen,

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

Sievennä:

$$\int{2 \sin{\left(\ln{\left(x \right)} \right)} d x} = - \sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}$$

Lisää integrointivakio:

$$\int{2 \sin{\left(\ln{\left(x \right)} \right)} d x} = - \sqrt{2} x \cos{\left(\ln{\left(x \right)} + \frac{\pi}{4} \right)}+C$$

$$$\int 2 \sin{\left(\ln\left(x\right) \right)}\, dx = - \sqrt{2} x \cos{\left(\ln\left(x\right) + \frac{\pi}{4} \right)} + C$$$A