Integralen av $$$\sqrt{2} \left(1 - \sin{\left(2 x \right)}\right)$$$

Bestäm $$$\int \sqrt{2} \left(1 - \sin{\left(2 x \right)}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\sqrt{2}$$$ och $$$f{\left(x \right)} = 1 - \sin{\left(2 x \right)}$$$:

$${\color{red}{\int{\sqrt{2} \left(1 - \sin{\left(2 x \right)}\right) d x}}} = {\color{red}{\sqrt{2} \int{\left(1 - \sin{\left(2 x \right)}\right)d x}}}$$

Integrera termvis:

$$\sqrt{2} {\color{red}{\int{\left(1 - \sin{\left(2 x \right)}\right)d x}}} = \sqrt{2} {\color{red}{\left(\int{1 d x} - \int{\sin{\left(2 x \right)} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\sqrt{2} \left(- \int{\sin{\left(2 x \right)} d x} + {\color{red}{\int{1 d x}}}\right) = \sqrt{2} \left(- \int{\sin{\left(2 x \right)} d x} + {\color{red}{x}}\right)$$

Låt $$$u=2 x$$$ vara.

Då $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{2}$$$.

Integralen kan omskrivas som

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

Integralen av sinus är $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

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

Kom ihåg att $$$u=2 x$$$:

$$\sqrt{2} \left(x + \frac{\cos{\left({\color{red}{u}} \right)}}{2}\right) = \sqrt{2} \left(x + \frac{\cos{\left({\color{red}{\left(2 x\right)}} \right)}}{2}\right)$$

Alltså,

$$\int{\sqrt{2} \left(1 - \sin{\left(2 x \right)}\right) d x} = \sqrt{2} \left(x + \frac{\cos{\left(2 x \right)}}{2}\right)$$

Lägg till integrationskonstanten:

$$\int{\sqrt{2} \left(1 - \sin{\left(2 x \right)}\right) d x} = \sqrt{2} \left(x + \frac{\cos{\left(2 x \right)}}{2}\right)+C$$

$$$\int \sqrt{2} \left(1 - \sin{\left(2 x \right)}\right)\, dx = \sqrt{2} \left(x + \frac{\cos{\left(2 x \right)}}{2}\right) + C$$$A