revising Anti derivitives, Cheat Sheet of Mathematics

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Antiderivatives

Discovery of Power Rule for

Antiderivatives

If f ‘ (x) =

Then f(x) =

If f ‘ (x) =

Then f(x) =

If f ‘ (x) =

Then f(x) =

If f ‘ (x) =

Then f(x) =

4 9 2 3

3 2 x  x  x 

x 3 x x 3 x

4 3 2   

x  3 x  2

x x 2 x

2

3

3

(^2 ) 2

3

 

4 5

3

2

3 2   x 

x

x

x x 2 x 5 x

2

3

5

(^3 ) 3

5

  



4 3 5

2  x  x 

x x 5 x

2

3

3

(^4 3 )  



Differentiation Integration

The process of finding

a derivative The process of finding

the antiderivative

Symbols: Symbols:

, y ', f '( x )

dx

dy



f ( x ) dx

Integral

Integrand

Tells us the

variable of

integration

f ( x ) dx is the indefinite integral of f(x) with respect to x.

Each function has more than one antiderivative (actually infinitely many)

3 2

3 2

3 2

3 2

x x

x x

x x

x x

Derivative of: 

The

antiderivatives

vary by a

constant!

Basic Integration Formulas

C

n

x

x dx

n n 







 1

1

kdx  kx  C 

C

a

ax

ax dx 



 

cos

sin( )

C

a

ax

ax dx   

sin

cos( )

C

a

ax

ax dx   

tan

sec ( )

2

C

a

ax

ax dx 



 

cot

csc ( )

2

C

a

ax

ax ax dx   

sec

(sec tan )

C

a

ax

ax ax dx 



 

csc

(csc cot )

Find:



x dx

5



dx

x



sin 2 xdx



dx

x

cos

C

x

 

6

6

 x dx  x  C  x  C 



2

1

2

1

C

x



cos 2

C

x

C

x

2 sin

sin

You can always check

your answer by

differentiating!

Evaluate:

  

5 x  x  6 x  4 dx

3 2

   

 5 x dx  x dx  6 xdx  4 dx

3 2

   

 5 x dx  x dx  6 xdx  4 dx

3 2

C x C

x

C

x

C

x

  

4 3 2

x x C

x x

   3  4 

4 3

4 3

C represents

any constant

Evaluate:



x dx

5 3



5

3

x C

x

 

5

8

 x  C

5

8

Evaluate:

  

 x dx

2 2 3

  

  x  x dx

2 4 9 6

C

x

 x  x  

5

5

3

x x C

x

  2  9 

5

3

5

Evaluate:



dx

x

x

2 cos

sin



  dx

x

x

x cos

sin

cos

sec x  C



 sec x tan xdx