Integralen av $$$\left(2 \sin{\left(\theta \right)} + 3\right)^{2}$$$

Bestäm $$$\int \left(2 \sin{\left(\theta \right)} + 3\right)^{2}\, d\theta$$$.

Expand the expression:

$${\color{red}{\int{\left(2 \sin{\left(\theta \right)} + 3\right)^{2} d \theta}}} = {\color{red}{\int{\left(4 \sin^{2}{\left(\theta \right)} + 12 \sin{\left(\theta \right)} + 9\right)d \theta}}}$$

Integrera termvis:

$${\color{red}{\int{\left(4 \sin^{2}{\left(\theta \right)} + 12 \sin{\left(\theta \right)} + 9\right)d \theta}}} = {\color{red}{\left(\int{9 d \theta} + \int{12 \sin{\left(\theta \right)} d \theta} + \int{4 \sin^{2}{\left(\theta \right)} d \theta}\right)}}$$

Tillämpa konstantregeln $$$\int c\, d\theta = c \theta$$$ med $$$c=9$$$:

$$\int{12 \sin{\left(\theta \right)} d \theta} + \int{4 \sin^{2}{\left(\theta \right)} d \theta} + {\color{red}{\int{9 d \theta}}} = \int{12 \sin{\left(\theta \right)} d \theta} + \int{4 \sin^{2}{\left(\theta \right)} d \theta} + {\color{red}{\left(9 \theta\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ med $$$c=4$$$ och $$$f{\left(\theta \right)} = \sin^{2}{\left(\theta \right)}$$$:

$$9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + {\color{red}{\int{4 \sin^{2}{\left(\theta \right)} d \theta}}} = 9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + {\color{red}{\left(4 \int{\sin^{2}{\left(\theta \right)} d \theta}\right)}}$$

Använd potensreduceringsformeln $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$ med $$$\alpha=\theta$$$:

$$9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 4 {\color{red}{\int{\sin^{2}{\left(\theta \right)} d \theta}}} = 9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 4 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 \theta \right)}}{2}\right)d \theta}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(\theta \right)} = 1 - \cos{\left(2 \theta \right)}$$$:

$$9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 4 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 \theta \right)}}{2}\right)d \theta}}} = 9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 4 {\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 \theta \right)}\right)d \theta}}{2}\right)}}$$

Integrera termvis:

$$9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 2 {\color{red}{\int{\left(1 - \cos{\left(2 \theta \right)}\right)d \theta}}} = 9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} + 2 {\color{red}{\left(\int{1 d \theta} - \int{\cos{\left(2 \theta \right)} d \theta}\right)}}$$

Tillämpa konstantregeln $$$\int c\, d\theta = c \theta$$$ med $$$c=1$$$:

$$9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 \int{\cos{\left(2 \theta \right)} d \theta} + 2 {\color{red}{\int{1 d \theta}}} = 9 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 \int{\cos{\left(2 \theta \right)} d \theta} + 2 {\color{red}{\theta}}$$

Låt $$$u=2 \theta$$$ vara.

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

Alltså,

$$11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 {\color{red}{\int{\cos{\left(2 \theta \right)} d \theta}}} = 11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

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

$$11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = 11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - 2 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - {\color{red}{\int{\cos{\left(u \right)} d u}}} = 11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - {\color{red}{\sin{\left(u \right)}}}$$

Kom ihåg att $$$u=2 \theta$$$:

$$11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - \sin{\left({\color{red}{u}} \right)} = 11 \theta + \int{12 \sin{\left(\theta \right)} d \theta} - \sin{\left({\color{red}{\left(2 \theta\right)}} \right)}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ med $$$c=12$$$ och $$$f{\left(\theta \right)} = \sin{\left(\theta \right)}$$$:

$$11 \theta - \sin{\left(2 \theta \right)} + {\color{red}{\int{12 \sin{\left(\theta \right)} d \theta}}} = 11 \theta - \sin{\left(2 \theta \right)} + {\color{red}{\left(12 \int{\sin{\left(\theta \right)} d \theta}\right)}}$$

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

$$11 \theta - \sin{\left(2 \theta \right)} + 12 {\color{red}{\int{\sin{\left(\theta \right)} d \theta}}} = 11 \theta - \sin{\left(2 \theta \right)} + 12 {\color{red}{\left(- \cos{\left(\theta \right)}\right)}}$$

Alltså,

$$\int{\left(2 \sin{\left(\theta \right)} + 3\right)^{2} d \theta} = 11 \theta - \sin{\left(2 \theta \right)} - 12 \cos{\left(\theta \right)}$$

Lägg till integrationskonstanten:

$$\int{\left(2 \sin{\left(\theta \right)} + 3\right)^{2} d \theta} = 11 \theta - \sin{\left(2 \theta \right)} - 12 \cos{\left(\theta \right)}+C$$

$$$\int \left(2 \sin{\left(\theta \right)} + 3\right)^{2}\, d\theta = \left(11 \theta - \sin{\left(2 \theta \right)} - 12 \cos{\left(\theta \right)}\right) + C$$$A