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11.4 How to find Antiderivatives

We can differentiate almost any function we can write down by our differentiation rules. There are noimmediate rules for antidifferentiation. 

We can however use what we know about differentiating, and the consequences of the differentiation rules to find antiderivatives for many kinds of functions.

We will study methods for doing this in further detail in a later section.

The most obvious method is that of working backwards: we know the antiderivative of functions that are derivatives of functions we know.We can therefore construct a list or table of antiderivatives by looking at a list of derivatives backwards.

We can also exploit the properties of derivatives to extend our list of antiderivatives.

The linear properties of the derivative: that the derivative of a constant multiple of a sum is the same multiple of the sum of the derivatives of the terms, translates to the same property for the  antiderivatives.

The chain rule for differentiating corresponds to the most important tool for extending our antiderivative table: it provides a method for reexpressing the antiderivative of f with respect to variable x as an antiderivative of a different function with respect to a new variable, u(x).

When either f or the new function has a known antiderivative, we can deduce the antiderivative of the other function.You should develop your own list of functions that you can recognize as those that you can find antiderivatives of.