The antiderivative (or indefinite integral
) is, essentially, a guess at the function from which a derivative
It is impossible to know the exact function due to the fact that the derivative of a constant is always . So, dy/dx(3x + 2) and dy/dx(3x + π) produce the same result- 3. In order to compensate, a constant of integration (usually c) is added to the antiderivative (ie: 3 + c).
Antiderivatives are helpful for finding the area under a curve (much more convenient than counting a bunch of rectangles) and rates of change among other things.
Primary methods of finding antiderivatives are as follows:Memorization: Yes, some people memorize integrals. Some that are common in memorization are:
∫sin(x) dx = -cos(x) + c
∫cos(x) dx = sin(x) + c
∫(1/cos2(x)) dx = ∫sec2(x) dx = tan(x) + c
u/du Substitution (like chunking): A value is substituted out of the integral in place of u (or any other variable) and the problem is re-written in terms of u and du (the derivative of u). When that is solved, the original value is plugged back in.
u/dv: A section of the problem is designated as u and the other as dv. The derivative of u (du or u') and the antiderivative of dv are calculated. Then, they are arranged as so:
u * v - ∫ du * v
and finally solved.
Triangles: When two things squared are subtracted or added, they can placed as legs (or with one as the hypotenuse) on a right triangle and reduced accordingly.