Integralen av $$$\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3}$$$

Bestäm $$$\int \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3}\, dx$$$.

Använd potensreduceringsformeln $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ med $$$\alpha=x$$$:

$${\color{red}{\int{\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3} d x}}} = {\color{red}{\int{\frac{\left(\cos{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{6} d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = \frac{\left(\cos{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{3}$$$:

$${\color{red}{\int{\frac{\left(\cos{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{6} d x}}} = {\color{red}{\left(\frac{\int{\frac{\left(\cos{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{3} d x}}{2}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\frac{\left(\cos{\left(2 x \right)} + 1\right) \sin{\left(x \right)}}{3} d x}}}}{2} = \frac{{\color{red}{\int{\left(\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} + \frac{\sin{\left(x \right)}}{3}\right)d x}}}}{2}$$

Integrera termvis:

$$\frac{{\color{red}{\int{\left(\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} + \frac{\sin{\left(x \right)}}{3}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x} + \int{\frac{\sin{\left(x \right)}}{3} d x}\right)}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$$\frac{\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x}}{2} + \frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{3} d x}}}}{2} = \frac{\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x}}{2} + \frac{{\color{red}{\left(\frac{\int{\sin{\left(x \right)} d x}}{3}\right)}}}{2}$$

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

$$\frac{\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x}}{2} + \frac{{\color{red}{\int{\sin{\left(x \right)} d x}}}}{6} = \frac{\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x}}{2} + \frac{{\color{red}{\left(- \cos{\left(x \right)}\right)}}}{6}$$

Skriv om $$$\sin\left(x \right)\cos\left(2 x \right)$$$ med hjälp av formeln $$$\sin\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \sin\left(\alpha-\beta \right)+\frac{1}{2} \sin\left(\alpha+\beta \right)$$$ med $$$\alpha=x$$$ och $$$\beta=2 x$$$:

$$- \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\int{\frac{\sin{\left(x \right)} \cos{\left(2 x \right)}}{3} d x}}}}{2} = - \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{6} + \frac{\sin{\left(3 x \right)}}{6}\right)d x}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = - \frac{\sin{\left(x \right)}}{3} + \frac{\sin{\left(3 x \right)}}{3}$$$:

$$- \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{6} + \frac{\sin{\left(3 x \right)}}{6}\right)d x}}}}{2} = - \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\left(\frac{\int{\left(- \frac{\sin{\left(x \right)}}{3} + \frac{\sin{\left(3 x \right)}}{3}\right)d x}}{2}\right)}}}{2}$$

Integrera termvis:

$$- \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{3} + \frac{\sin{\left(3 x \right)}}{3}\right)d x}}}}{4} = - \frac{\cos{\left(x \right)}}{6} + \frac{{\color{red}{\left(- \int{\frac{\sin{\left(x \right)}}{3} d x} + \int{\frac{\sin{\left(3 x \right)}}{3} d x}\right)}}}{4}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$$- \frac{\cos{\left(x \right)}}{6} + \frac{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}{4} - \frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{3} d x}}}}{4} = - \frac{\cos{\left(x \right)}}{6} + \frac{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}{4} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(x \right)} d x}}{3}\right)}}}{4}$$

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

$$- \frac{\cos{\left(x \right)}}{6} + \frac{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}{4} - \frac{{\color{red}{\int{\sin{\left(x \right)} d x}}}}{12} = - \frac{\cos{\left(x \right)}}{6} + \frac{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}{4} - \frac{{\color{red}{\left(- \cos{\left(x \right)}\right)}}}{12}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(x \right)} = \sin{\left(3 x \right)}$$$:

$$- \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\int{\frac{\sin{\left(3 x \right)}}{3} d x}}}}{4} = - \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\left(\frac{\int{\sin{\left(3 x \right)} d x}}{3}\right)}}}{4}$$

Låt $$$u=3 x$$$ vara.

Då $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{3}$$$.

Alltså,

$$- \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\int{\sin{\left(3 x \right)} d x}}}}{12} = - \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{12}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{3}$$$ och $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$- \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{3} d u}}}}{12} = - \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{3}\right)}}}{12}$$

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

$$- \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{36} = - \frac{\cos{\left(x \right)}}{12} + \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{36}$$

Kom ihåg att $$$u=3 x$$$:

$$- \frac{\cos{\left(x \right)}}{12} - \frac{\cos{\left({\color{red}{u}} \right)}}{36} = - \frac{\cos{\left(x \right)}}{12} - \frac{\cos{\left({\color{red}{\left(3 x\right)}} \right)}}{36}$$

Alltså,

$$\int{\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3} d x} = - \frac{\cos{\left(x \right)}}{12} - \frac{\cos{\left(3 x \right)}}{36}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3} d x} = - \frac{\cos{\left(x \right)}}{12} - \frac{\cos{\left(3 x \right)}}{36}+C$$

$$$\int \frac{\sin{\left(x \right)} \cos^{2}{\left(x \right)}}{3}\, dx = \left(- \frac{\cos{\left(x \right)}}{12} - \frac{\cos{\left(3 x \right)}}{36}\right) + C$$$A