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Lecture 1

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

1This

**preview**shows half of the first page. to view the full**3 pages of the document.** MATH 2B - Lecture 1 - Anti-derivatives

Anti-differentiation is exactly what it sounds like: the opposite of differentiation. That is,

given a function f, can we nd a function F whose derivative is f.

Denition. An anti-derivative of a function f is a function F such that F’(x) = f(x) for all x.

Examples

1. is an anti-derivative of (x) sin xF = (x)os xf =c

2. is an anti-derivative of (x)x)F= ( 2+ 3 3/2 (x) 3xf = √x2+ 3

3. is an anti-derivative of (x)F=x−3

2

cos x(x)f=cos x

2x+(x−3)tan x

2

.

We can easily check the veracity of these statements by differentiating F(x). But what if

you are asked to nd an anti-derivative, not just check something you’ve been given? In

general this is a hard problem.

● Differentiation is often described as easy in the sense that nice functions can

usually be differentiated via familiar rules

● Anti-differentiation is hard: very few functions have anti-derivatives that can

easily be computed

○ Example

: It is easy to find the derivative of , but can you find an(x)f=e−x2

anti-derivative of f(x)?

The only method that really exists for explicitly computing anti-derivatives is guess and

differentiate! Every famous rule that you’ll study in Integration (substitution, integration

by parts, etc.) is merely the result of guessing a general anti-derivative and checking

that your guess is correct

Anti-derivatives of Common Functions

● Guessing is, of course, easier if you have familiarity with differentiation. With

each of the following functions f(x) you should be able to guess the chosen

anti-derivative F(x) just from what you know about derivatives

Examples

1. has an anti-derivative of (x)f= 3 + x

2(x)x ln|x|f= 3 + 2

2. has an anti-derivative (x)sec x ef = 3 2− 2 3x(x)tan x ef = 3 − 3

23x

3. Find an anti-derivative of (x)os xg =c+x2

x+

3√x

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