Integralen av $$$5 \cos{\left(x \right)} \cos{\left(5 x \right)}$$$

Bestäm $$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx$$$.

Skriv om $$$\cos\left(x \right)\cos\left(5 x \right)$$$ med hjälp av formeln $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ med $$$\alpha=x$$$ och $$$\beta=5 x$$$:

$${\color{red}{\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x}}} = {\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = 5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}$$$:

$${\color{red}{\int{\left(\frac{5 \cos{\left(4 x \right)}}{2} + \frac{5 \cos{\left(6 x \right)}}{2}\right)d x}}} = {\color{red}{\left(\frac{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}{2}\right)}}$$

Integrera termvis:

$$\frac{{\color{red}{\int{\left(5 \cos{\left(4 x \right)} + 5 \cos{\left(6 x \right)}\right)d x}}}}{2} = \frac{{\color{red}{\left(\int{5 \cos{\left(4 x \right)} d x} + \int{5 \cos{\left(6 x \right)} d x}\right)}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\int{5 \cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{{\color{red}{\left(5 \int{\cos{\left(4 x \right)} d x}\right)}}}{2}$$

Låt $$$u=4 x$$$ vara.

Då $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{4}$$$.

Integralen blir

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{4}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2}$$

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

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{8}$$

Kom ihåg att $$$u=4 x$$$:

$$\frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{8} = \frac{\int{5 \cos{\left(6 x \right)} d x}}{2} + \frac{5 \sin{\left({\color{red}{\left(4 x\right)}} \right)}}{8}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = \cos{\left(6 x \right)}$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\int{5 \cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{{\color{red}{\left(5 \int{\cos{\left(6 x \right)} d x}\right)}}}{2}$$

Låt $$$u=6 x$$$ vara.

Då $$$du=\left(6 x\right)^{\prime }dx = 6 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{6}$$$.

Alltså,

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{6}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{2} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{2}$$

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

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 {\color{red}{\sin{\left(u \right)}}}}{12}$$

Kom ihåg att $$$u=6 x$$$:

$$\frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{u}} \right)}}{12} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left({\color{red}{\left(6 x\right)}} \right)}}{12}$$

Alltså,

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \sin{\left(4 x \right)}}{8} + \frac{5 \sin{\left(6 x \right)}}{12}$$

Förenkla:

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}$$

Lägg till integrationskonstanten:

$$\int{5 \cos{\left(x \right)} \cos{\left(5 x \right)} d x} = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24}+C$$

$$$\int 5 \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx = \frac{5 \left(3 \sin{\left(4 x \right)} + 2 \sin{\left(6 x \right)}\right)}{24} + C$$$A