## Calculus 8th Edition

$2sin(x)+c$
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: $2sin(x)+2c= 2sin(x)+c$