Calculus of a Single Variable, Chapter 6, 6.3, Section 6.3, Problem 9

The general solution of a differential equation in a form of y' = f(x) can
be evaluated using direct integration. The derivative of y denoted as y' can be written as (dy)/(dx) then y'= f(x) can be expressed as (dy)/(dx)= f(x) .
For the problem yy'=4sin(x) , we may apply y' = (dy)/(dx) to set-up the integration:
y(dy)/(dx)= 4sin(x) .
or y dy = 4 sin(x) dx

Then set-up direct integration on both sides:
inty dy = int 4 sin(x) dx
Integration:
Apply Power Rule integration: int u^n du= u^(n+1)/(n+1) on inty dy .
Note: y is the same as y^1 .
int y dy = y^(1+1)/(1+1)
= y^2/2
Apply the basic integration property: int c*f(x)dx= c int f(x) dx and basic integration formula for sine function: int sin(u) du = -cos(u) +C
int 4 sin(x) dx= 4int sin(x) dx
= -4 cos(x) +C

Then combining the results for the general solution of differential equation:
y^2/2 = -4cos(x)+C
2* [y^2/2] = 2*[-4cos(x)]+C
y^2 =-8cos(x)+C
y = +-sqrt(C-8cosx)