Answer:
[tex] \displaystyle \rm x \cos(a) + \sin(a) \ln\left( | \sin(x) | \right) + C[/tex]
Step-by-step explanation:
we would like to integrate the following integration:
[tex] \displaystyle \int \frac{ \sin(x + a) }{ \sin(x) } dx[/tex]
we can rewrite the denominator by using algebraic identity given by:
[tex] \displaystyle \rm\sin( \alpha \pm \beta ) = \sin( \alpha ) \cos( \beta ) \pm \cos( \alpha ) \sin( \beta ) [/tex]
thus substitute:
[tex] \displaystyle \int \frac{ \sin(x ) \cos(a) + \cos(x) \sin(a) } { \sin(x) } dx[/tex]
we should rewrite integrand as sum therefore we can use sum integration formula
[tex] \displaystyle \rm\int \frac{ \sin(x ) \cos(a) } { \sin(x) } + \frac{ \cos(x) \sin( \alpha ) } { \sin(x) } dx[/tex]
use sum integration formula:
[tex] \displaystyle \rm\int \frac{ \sin(x ) \cos(a) } { \sin(x) } dx + \int \frac{ \cos(x) \sin( \alpha ) } { \sin(x) } dx[/tex]
reduce fraction:
[tex] \displaystyle \rm\int \cos(a) dx + \int \frac{ \cos(x) \sin( \alpha ) } { \sin(x) } dx[/tex]
rewrite:
[tex] \displaystyle \rm\int \cos(a) dx + \int \frac{ \cos(x) } { \sin(x) } \cdot \sin( a) dx[/tex]
use trigonometric indentity:
[tex] \displaystyle \rm\int \cos(a) dx + \int \cot(x) \cdot \sin( a) dx[/tex]
use constant integration formula
[tex] \displaystyle \rm\int \cos(a) dx + \sin(a) \int \cot(x) dx[/tex]
use integration rules:
[tex] \displaystyle \rm x \cos(a) + \sin(a) \ln\left( | \sin(x) | \right)[/tex]
and finally we of course have to add constant of integration
[tex] \displaystyle \rm x \cos(a) + \sin(a) \ln\left( | \sin(x) | \right) + C[/tex]
And we are done!
[tex]\text{Note:I used integration by substitution to figure out }\\\displaystyle \int \cot(x)dx \:\text{part}[/tex]
