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Antiderivatives and Indefinite Integrals- FTC1

Objectives

  • Present a table of basic antiderivatives.

Summary

To find algebraic formulas for antiderivatives of more complicated algebraic functions, we need to think carefully about how we can reverse known differentiation rules. To that end, it is essential that we understand and recall known derivatives of basic functions, as well as the standard derivative rules.

The indefinite integral provides notation for antiderivatives. When we write " ∫ f ( x ) d x ", we mean "the general antiderivative of f ". In particular, if we have functions f and F such that F ′ = f , the following two statements say the exact thing:

d d x [ F ( x ) ] = f ( x )   and ∫ f ( x ) d x = F ( x ) + C That is, f is the derivative of F , and F is an antiderivative of f .

Rules

∫ x n d x = x n + 1 n + C
∫ 1 a x d x = l n ( a x ) + C
∫ c o s x d x = s i n x + C
∫ s i n x d x = − c o s x + C
∫ s e c 2 x d x = t a n x + C
∫ s e c x t a n x d x = s e c x + C
∫ e a x d x = e a x a + C
Please refer to the video below for more examples on the applications on the rules above as well as examples of Integration By Substitution and Integration By Part.

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