..what to do to create the magic

If the integrand is cot 2 x or tan 2 x…
Use the Pythagorean identities
Example:
 tan xdx 

If the integrand is a fraction and… the fraction is top heavy or equal degree…
Divide and separate
Example:

If the integrand is a fraction and… the fraction is top heavy or equal degree…
Divide and separate
Example:

u-substitution with a twist…
 x x

5 dx


 u-substitution with a twist…
 3


Continuous Piecewise functions


3x

2  4x 2,x 0



4
2 f
  dx

2


1 f
  dx


Continuous Piecewise functions

 

 e

0 f
  dx


Use the substitution method to determine
 tan xdx
 cot u du

Summary of trig Integrals you didn’t already know…
 tan u du
  ln | cos |
  ln | sec |

C
 cot u du
 ln | sin |

C
 sec u du
 ln | sec u
 u

C
 csc u du
  ln | csc u
 u

C

 csc(3 )



Evaluate sec x
  dx


Evaluate
 x tan( x
2
) dx


Evaluate


2

3


Substitution (changing the bounds)
1

0 x
 x
2 
2

3 dx


AP Multiple Choice
If the substitution u = 1 – 2x 4 is used which of the following is equivalent to
2

0 x
3
1

2 x
4 dx
A )
D )

1
8
2

0 u

1
2 du

1
8
1


31 u

1
2 du
B )
E )
1
4
2

0 u

1
2 du

1
4
1

31
 u

1
2 du
C )

8
2

0 u

1
2 du
F )
1


31 u

1
2 du