Answer
$2sin(x)+c$
Work Step by Step
Given $\int \dfrac{sin(2x)}{sin(x)}dx$
1.) Using the double angle identitie: $sin(2x)=2sin(x)cos(x) , $ we have:
$\int \dfrac{sin(2x)}{sin(x)}dx = \int \dfrac{2sin(x)cos(x)}{sin(x)}dx \\$
2.) Simplifyng:
$\int \dfrac{2sin(x)cos(x)}{sin(x)}dx = \int 2cos(x)dx \\$
3.) Taking out the constant multiple:
$ \int 2cos(x)dx = 2\int cos(x)dx \\$
4.) Solving the integral:
$ 2\int cos(x)dx= 2(sin(x)+c) \\$
5.) Performing the distributive property:
$2(sin(x)+c)= 2sin(x)+2c \\$
6.) Since a constant ($c$) times $2$ it's still a constant:
$2sin(x)+2c= 2sin(x)+c$
