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

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

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

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

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 $$$f{\left(t \right)} = \left(1 - \cos{\left(2 t \right)}\right) \cos{\left(2 t \right)}$$$:

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

Expand the expression:

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

Integrate term by term:

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

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

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

The integral can be rewritten as

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

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 $$$f{\left(u \right)} = \cos^{2}{\left(u \right)}$$$:

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

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

$$\frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}}{4} = \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{4}$$

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 $$$f{\left(u \right)} = \cos{\left(2 u \right)} + 1$$$:

$$\frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{4} = \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}}{4}$$

Integrate term by term:

$$\frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}}}{8} = \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\left(\int{1 d u} + \int{\cos{\left(2 u \right)} d u}\right)}}}{8}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$\frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\int{\cos{\left(2 u \right)} d u}}{8} - \frac{{\color{red}{\int{1 d u}}}}{8} = \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\int{\cos{\left(2 u \right)} d u}}{8} - \frac{{\color{red}{u}}}{8}$$

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,

$$- \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{8} = - \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{8}$$

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 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$:

$$- \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{8} = - \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{8}$$

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

$$- \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{16} = - \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{{\color{red}{\sin{\left(v \right)}}}}{16}$$

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

$$- \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\sin{\left({\color{red}{v}} \right)}}{16} = - \frac{u}{8} + \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{16}$$

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

$$\frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\sin{\left(2 {\color{red}{u}} \right)}}{16} - \frac{{\color{red}{u}}}{8} = \frac{\int{\cos{\left(2 t \right)} d t}}{2} - \frac{\sin{\left(2 {\color{red}{\left(2 t\right)}} \right)}}{16} - \frac{{\color{red}{\left(2 t\right)}}}{8}$$

The integral $$$\int{\cos{\left(2 t \right)} d t}$$$ was already calculated:

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

Therefore,

$$- \frac{t}{4} - \frac{\sin{\left(4 t \right)}}{16} + \frac{{\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{2} = - \frac{t}{4} - \frac{\sin{\left(4 t \right)}}{16} + \frac{{\color{red}{\left(\frac{\sin{\left(2 t \right)}}{2}\right)}}}{2}$$

Therefore,

$$\int{\sin^{2}{\left(t \right)} \cos{\left(2 t \right)} d t} = - \frac{t}{4} + \frac{\sin{\left(2 t \right)}}{4} - \frac{\sin{\left(4 t \right)}}{16}$$

Add the constant of integration:

$$\int{\sin^{2}{\left(t \right)} \cos{\left(2 t \right)} d t} = - \frac{t}{4} + \frac{\sin{\left(2 t \right)}}{4} - \frac{\sin{\left(4 t \right)}}{16}+C$$

$$$\int \sin^{2}{\left(t \right)} \cos{\left(2 t \right)}\, dt = \left(- \frac{t}{4} + \frac{\sin{\left(2 t \right)}}{4} - \frac{\sin{\left(4 t \right)}}{16}\right) + C$$$A