Integral of $$$\frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}}$$$ with respect to $$$x$$$

Find $$$\int \frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{\sin{\left(\frac{\pi t}{4} \right)}}$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}} d x}}} = {\color{red}{\frac{\int{\sin{\left(x \right)} d x}}{\sin{\left(\frac{\pi t}{4} \right)}}}}$$

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

$$\frac{{\color{red}{\int{\sin{\left(x \right)} d x}}}}{\sin{\left(\frac{\pi t}{4} \right)}} = \frac{{\color{red}{\left(- \cos{\left(x \right)}\right)}}}{\sin{\left(\frac{\pi t}{4} \right)}}$$

Therefore,

$$\int{\frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}} d x} = - \frac{\cos{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}}$$

Add the constant of integration:

$$\int{\frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}} d x} = - \frac{\cos{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}}+C$$

$$$\int \frac{\sin{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}}\, dx = - \frac{\cos{\left(x \right)}}{\sin{\left(\frac{\pi t}{4} \right)}} + C$$$A