Integral of $$$4 \cos^{4}{\left(\frac{\theta}{2} \right)}$$$

Find $$$\int 4 \cos^{4}{\left(\frac{\theta}{2} \right)}\, d\theta$$$.

Apply the constant multiple rule $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ with $$$c=4$$$ and $$$f{\left(\theta \right)} = \cos^{4}{\left(\frac{\theta}{2} \right)}$$$:

$${\color{red}{\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}}} = {\color{red}{\left(4 \int{\cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}\right)}}$$

Let $$$u=\frac{\theta}{2}$$$.

Then $$$du=\left(\frac{\theta}{2}\right)^{\prime }d\theta = \frac{d\theta}{2}$$$ (steps can be seen »), and we have that $$$d\theta = 2 du$$$.

So,

$$4 {\color{red}{\int{\cos^{4}{\left(\frac{\theta}{2} \right)} d \theta}}} = 4 {\color{red}{\int{2 \cos^{4}{\left(u \right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=2$$$ and $$$f{\left(u \right)} = \cos^{4}{\left(u \right)}$$$:

$$4 {\color{red}{\int{2 \cos^{4}{\left(u \right)} d u}}} = 4 {\color{red}{\left(2 \int{\cos^{4}{\left(u \right)} d u}\right)}}$$

Apply the power reducing formula $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$ with $$$\alpha= u $$$:

$$8 {\color{red}{\int{\cos^{4}{\left(u \right)} d u}}} = 8 {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{8}$$$ and $$$f{\left(u \right)} = 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3$$$:

$$8 {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}} = 8 {\color{red}{\left(\frac{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}{8}\right)}}$$

Integrate term by term:

$${\color{red}{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}} = {\color{red}{\left(\int{3 d u} + \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=3$$$:

$$\int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u} + {\color{red}{\int{3 d u}}} = \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u} + {\color{red}{\left(3 u\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=4$$$ and $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$:

$$3 u + \int{\cos{\left(4 u \right)} d u} + {\color{red}{\int{4 \cos{\left(2 u \right)} d u}}} = 3 u + \int{\cos{\left(4 u \right)} d u} + {\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}$$

Let $$$v=2 u$$$.

Then $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (steps can be seen »), and we have that $$$du = \frac{dv}{2}$$$.

Thus,

$$3 u + \int{\cos{\left(4 u \right)} d u} + 4 {\color{red}{\int{\cos{\left(2 u \right)} d u}}} = 3 u + \int{\cos{\left(4 u \right)} d u} + 4 {\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

$$3 u + \int{\cos{\left(4 u \right)} d u} + 4 {\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}} = 3 u + \int{\cos{\left(4 u \right)} d u} + 4 {\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}$$

The integral of the cosine is $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$$3 u + \int{\cos{\left(4 u \right)} d u} + 2 {\color{red}{\int{\cos{\left(v \right)} d v}}} = 3 u + \int{\cos{\left(4 u \right)} d u} + 2 {\color{red}{\sin{\left(v \right)}}}$$

Recall that $$$v=2 u$$$:

$$3 u + \int{\cos{\left(4 u \right)} d u} + 2 \sin{\left({\color{red}{v}} \right)} = 3 u + \int{\cos{\left(4 u \right)} d u} + 2 \sin{\left({\color{red}{\left(2 u\right)}} \right)}$$

Let $$$v=4 u$$$.

Then $$$dv=\left(4 u\right)^{\prime }du = 4 du$$$ (steps can be seen »), and we have that $$$du = \frac{dv}{4}$$$.

Therefore,

$$3 u + 2 \sin{\left(2 u \right)} + {\color{red}{\int{\cos{\left(4 u \right)} d u}}} = 3 u + 2 \sin{\left(2 u \right)} + {\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

$$3 u + 2 \sin{\left(2 u \right)} + {\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}} = 3 u + 2 \sin{\left(2 u \right)} + {\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}$$

The integral of the cosine is $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$$3 u + 2 \sin{\left(2 u \right)} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{4} = 3 u + 2 \sin{\left(2 u \right)} + \frac{{\color{red}{\sin{\left(v \right)}}}}{4}$$

Recall that $$$v=4 u$$$:

$$3 u + 2 \sin{\left(2 u \right)} + \frac{\sin{\left({\color{red}{v}} \right)}}{4} = 3 u + 2 \sin{\left(2 u \right)} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{4}$$

Recall that $$$u=\frac{\theta}{2}$$$:

$$2 \sin{\left(2 {\color{red}{u}} \right)} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{4} + 3 {\color{red}{u}} = 2 \sin{\left(2 {\color{red}{\left(\frac{\theta}{2}\right)}} \right)} + \frac{\sin{\left(4 {\color{red}{\left(\frac{\theta}{2}\right)}} \right)}}{4} + 3 {\color{red}{\left(\frac{\theta}{2}\right)}}$$

Therefore,

$$\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta} = \frac{3 \theta}{2} + 2 \sin{\left(\theta \right)} + \frac{\sin{\left(2 \theta \right)}}{4}$$

Simplify:

$$\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta} = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4}$$

Add the constant of integration:

$$\int{4 \cos^{4}{\left(\frac{\theta}{2} \right)} d \theta} = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4}+C$$

$$$\int 4 \cos^{4}{\left(\frac{\theta}{2} \right)}\, d\theta = \frac{6 \theta + 8 \sin{\left(\theta \right)} + \sin{\left(2 \theta \right)}}{4} + C$$$A