1

Lecture 9 – Integration BasicsFunctions – know their shapes and properties

xexf )(

A few (very few) examples:

xxf ln)(

xxf 1sin)(

1 1

2

xxf 1tan)(

2

Trigonometric RulesKnow basics about sine, cosine, tangent, secant, plus 2 right triangles

Beyond these angles:

and use reference angles for all quadrants.

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Substitution RuleFirst approach for any integral should be a u-substitution.

Ex. 1 Which (if any) of the following can use a basic u-substitution?

41 t

dt 41

t

dtt 4

2

1

t

dtt 4

3

1

t

dtt

4

Ex. 2 Which (if any) of the following can use a basic u-substitution?

dxxx )sin( dxxx )sin(2 dxxx )sin( 2

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Know derivatives for trig functions.

Ex. 3 But what about antiderivatives?

dxx tan

xxyxy

xyxy

xyxy

xyx

tansec',sec

sec',tan

sin',cos

cos',siny

2

Cxdxx

Cxdxx

sin cos

cos sin

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Ex.4

What antiderivative for secant function?

dxx sec

xxyxy

xyxy

tansec',sec

sec',tan 2

7

1 xe

dxEx.5

8

1 xe

dx

Need to try a different 1.