MATH 155/MYERS/ALL FORMULAS ON THIS HANDOUT MUST BE MEMORIZED! INTEGRAL REVIEW
THEOREM: THE CONSTANT RULE Let k be a real number.
kdx x C �³
Example 1: Find the indefinite integral.
3dx�³
THEOREM: THE POWER RULE Let n be a rational number.
1
, 11
nn xx dx C n
n
�
� z ��³
Example 2: Find the following indefinite integrals.
a. 5x dx�³
K
1 3 7
to
2
b. 1 2x dx³
THEOREM: THE CONSTANT MULTIPLE RULE If f is an integrable function and c is a real number, then cf is also integrable and
� � � �cf x dx c f x dx ³ ³
Example 3: Find the area of the region bounded by � � 32f x x , 1x , 3x ,
and 0y .
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THEOREM: THE SUM AND DIFFERENCE RULES The sum (or difference) of two integrable functions f and g is itself integrable. Moreover, the antiderivative of f g� (or f g� ) is the sum (or difference) of the antiderivatives of f and g .
� � � � � � � �f x g x dx f x dx g x dx� �ª º¬ ¼³ ³ ³
� � � � � � � �f x g x dx f x dx g x dx� �ª º¬ ¼³ ³ ³
Example 4: Find the indefinite integral.
a. 25x x dx
x§ ·�¨ ¸¨ ¸© ¹³
b. � �23 1x dx�³
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THEOREM: ANTIDERIVATIVES OF THE TRIGONOMETRIC FUNCTIONS
2 2
sin cos cos sin
csc cot csc sec tan sec
sec tan csc cot
tan ln cos cot ln sin
sec ln sec tan
xdx x C xdx x C
x xdx x C x xdx x C
xdx x C xdx x C
xdx x C xdx x C
xdx x x
� � �
� � �
� � �
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�
³ ³³ ³³ ³³ ³³ csc ln csc cotC xdx x x C� � � �³
Example 5: Integrate.
a. 2sec xdx³
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c. 3tan xdx³
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THEOREM: ANTIDIFFERENTIATION OF A COMPOSITE FUNCTION Let g be a function whose range is an interval I and let f be a function that is continuous on I . If g is differentiable on its domain and F is an antiderivative of f on I , then
� �� � � � � �� �f g x g x dx F g x Cc �³
Letting � �u g x gives � �du g x dxc and
� � � �f u du F u C �³
Example 6: Find the following definite and indefinite integrals.
a. � �1x x dx�³
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c. 2cos 3xdx³
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Theorem: LOG RULE FOR INTEGRATION
Let u be a differentiable function of x.
1 11. ln 2. lndx x C du u Cx u
� �³ ³
Theorem: INTEGRATION RULES FOR EXPONENTIAL FUNCTIONS
Let u be a differentiable function of x.
1. 2.
13. , is a positive real number, 1ln
x x u u
x x
e dx e C e du e C
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Example 7: Find the following indefinite integrals and evaluate the definite integrals.
a. 25 12
t t dxt
� ��³
b. � �25
lndx
x x³
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Theorem: INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
Let u be a differentiable function of x , and let 0a! .
2 2arcsindu u C
aa u �
�³
2 2
1 arcsecudu C
a au u a �
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��³
Example 7: Find the following indefinite integrals and evaluate the definite integrals.
a. 4 2t dt
t �³
sin wz etc
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