Integral of $$$\cos^{2}{\left(t \right)}$$$

Find $$$\int \cos^{2}{\left(t \right)}\, dt$$$.

Apply the power reducing formula $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ with $$$\alpha=t$$$:

$${\color{red}{\int{\cos^{2}{\left(t \right)} d t}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)d t}}}$$

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

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dt = c t$$$ with $$$c=1$$$:

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

Let $$$u=2 t$$$.

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

Therefore,

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

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

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

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

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

Recall that $$$u=2 t$$$:

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

Therefore,

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

Add the constant of integration:

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

$$$\int \cos^{2}{\left(t \right)}\, dt = \left(\frac{t}{2} + \frac{\sin{\left(2 t \right)}}{4}\right) + C$$$A