Integral of $$$\left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)}$$$

Find $$$\int \left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)}\, dt$$$.

Let $$$u=1 - \sin{\left(\frac{t}{2} \right)}$$$.

Then $$$du=\left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{\prime }dt = - \frac{\cos{\left(\frac{t}{2} \right)}}{2} dt$$$ (steps can be seen »), and we have that $$$\cos{\left(\frac{t}{2} \right)} dt = - 2 du$$$.

Thus,

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

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

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

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- 2 {\color{red}{\int{u^{2} d u}}}=- 2 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- 2 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Recall that $$$u=1 - \sin{\left(\frac{t}{2} \right)}$$$:

$$- \frac{2 {\color{red}{u}}^{3}}{3} = - \frac{2 {\color{red}{\left(1 - \sin{\left(\frac{t}{2} \right)}\right)}}^{3}}{3}$$

Therefore,

$$\int{\left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)} d t} = - \frac{2 \left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{3}}{3}$$

Simplify:

$$\int{\left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)} d t} = \frac{2 \left(\sin{\left(\frac{t}{2} \right)} - 1\right)^{3}}{3}$$

Add the constant of integration:

$$\int{\left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)} d t} = \frac{2 \left(\sin{\left(\frac{t}{2} \right)} - 1\right)^{3}}{3}+C$$

$$$\int \left(1 - \sin{\left(\frac{t}{2} \right)}\right)^{2} \cos{\left(\frac{t}{2} \right)}\, dt = \frac{2 \left(\sin{\left(\frac{t}{2} \right)} - 1\right)^{3}}{3} + C$$$A