The antiderivative (or indefinite

integral) is, essentially, a guess at the function from which a

derivative came.

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/cos^{2}(x)) *dx* = ∫sec^{2}(x) *dx* = tan(x) + c

Among others..

*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.