Integral of $$$3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1$$$

Find $$$\int \left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)\, dt$$$.

Integrate term by term:

$${\color{red}{\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t}}} = {\color{red}{\left(\int{1 d t} + \int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t}\right)}}$$

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

$$\int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{1 d t}}} = \int{18 \sin{\left(t \right)} d t} + \int{3 \sin^{2}{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{t}}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=3$$$ and $$$f{\left(t \right)} = \sin^{2}{\left(t \right)}$$$:

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{3 \sin^{2}{\left(t \right)} d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\left(3 \int{\sin^{2}{\left(t \right)} d t}\right)}}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\sin^{2}{\left(t \right)} d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right)d t}}}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\int{\left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right)d t}}} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 3 {\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 t \right)}\right)d t}}{2}\right)}}$$

Integrate term by term:

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + \frac{3 {\color{red}{\int{\left(1 - \cos{\left(2 t \right)}\right)d t}}}}{2} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + \frac{3 {\color{red}{\left(\int{1 d t} - \int{\cos{\left(2 t \right)} d t}\right)}}}{2}$$

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

$$t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \int{\cos{\left(2 t \right)} d t}}{2} + \frac{3 {\color{red}{\int{1 d t}}}}{2} = t + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \int{\cos{\left(2 t \right)} d t}}{2} + \frac{3 {\color{red}{t}}}{2}$$

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,

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\cos{\left(2 t \right)} d t}}}}{2} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2}$$

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

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{2} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{2}$$

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

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{4} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 {\color{red}{\sin{\left(u \right)}}}}{4}$$

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

$$\frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \sin{\left({\color{red}{u}} \right)}}{4} = \frac{5 t}{2} + \int{18 \sin{\left(t \right)} d t} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} - \frac{3 \sin{\left({\color{red}{\left(2 t\right)}} \right)}}{4}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=18$$$ and $$$f{\left(t \right)} = \sin{\left(t \right)}$$$:

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\int{18 \sin{\left(t \right)} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + {\color{red}{\left(18 \int{\sin{\left(t \right)} d t}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(t \right)} d t} = - \cos{\left(t \right)}$$$:

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 18 {\color{red}{\int{\sin{\left(t \right)} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} + \int{\frac{86 \cos{\left(t \right)}}{21} d t} + 18 {\color{red}{\left(- \cos{\left(t \right)}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=\frac{86}{21}$$$ and $$$f{\left(t \right)} = \cos{\left(t \right)}$$$:

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + {\color{red}{\int{\frac{86 \cos{\left(t \right)}}{21} d t}}} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + {\color{red}{\left(\frac{86 \int{\cos{\left(t \right)} d t}}{21}\right)}}$$

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

$$\frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + \frac{86 {\color{red}{\int{\cos{\left(t \right)} d t}}}}{21} = \frac{5 t}{2} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)} + \frac{86 {\color{red}{\sin{\left(t \right)}}}}{21}$$

Therefore,

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{5 t}{2} + \frac{86 \sin{\left(t \right)}}{21} - \frac{3 \sin{\left(2 t \right)}}{4} - 18 \cos{\left(t \right)}$$

Simplify:

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84}$$

Add the constant of integration:

$$\int{\left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)d t} = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84}+C$$

$$$\int \left(3 \sin^{2}{\left(t \right)} + 18 \sin{\left(t \right)} + \frac{86 \cos{\left(t \right)}}{21} + 1\right)\, dt = \frac{210 t + 344 \sin{\left(t \right)} - 63 \sin{\left(2 t \right)} - 1512 \cos{\left(t \right)}}{84} + C$$$A