direct integration
Direct integration is the most basic way to undo a derivative: if you know y' you find y simply by integrating, adding an arbitrary constant for the information the derivative threw away. It is the single move at the heart of the easiest ODEs, and it is the same antidifferentiation you met in first-semester calculus, now seen as solving an equation.
The mechanism is exactly the fundamental theorem of calculus run in reverse. Given y' = f(x), you compute an antiderivative F(x) with F'(x) = f(x), and then every solution is y = F(x) + C. The arbitrary constant C appears because differentiating wipes out constants — infinitely many curves have the same slope function, differing only by a vertical shift, and C labels which one you mean. If you also have an initial condition y(x_0) = y_0, you substitute it to solve for C and select the single curve through that point. You can also write the answer with a definite integral, y(x) = y_0 + integral from x_0 to x of f(s) ds, which builds the initial condition in automatically.
Direct integration is the workhorse behind pure-time equations and the final step of nearly every other method — separation ends in two integrations, the integrating-factor recipe ends in one. The honest limitation is inherited from calculus itself: many perfectly innocent functions have no antiderivative expressible in elementary terms (e^(-x^2), sin(x)/x, e^x/x), so 'solve by direct integration' sometimes means 'leave the answer as an integral' or evaluate it numerically. The equation is solved; the integral may simply have no closed form.
Solve y' = cos x + 4x^3. Antidifferentiate term by term: y = sin x + x^4 + C. If y(0) = 5, then sin 0 + 0 + C = 5 gives C = 5, so y = sin x + x^4 + 5.
Antidifferentiate, attach the constant C, then fix C from the initial condition.
Never forget the +C: dropping it gives only one curve out of an entire family, and you would be unable to satisfy a general initial condition.