Chapter 5
Integration
5.1 Anti-Derivatives
Many operations in mathematics have reverses – compare addition and subtraction,
multiplication and division, and powers and roots. We now know how to find the
derivatives of many functions. The reverse operation, anti-derivative will receive our
attention in this chapter. A function is an anti-derivative (or indefinite integral) of
if
= ,
we call the anti-derivative (or indefinite integral) of . For example, if ( ) = , we
can find its anti-derivative by realizing that for the function ( ) =
1 1
= = .2 = = ( ).
2 2
Thus ( ) = is an anti-derivative of ( ) = .
However, if is a constant:
1 1
+ = .2 = .
2 2
Since the derivative of a constant is zero. The family of all anti-derivatives of is thus
+ , where can be any constant.
Note that you should always check an anti-derivative by differentiating it and seeing
that you recover .
5.2 Indefinite integral
We use the notation ∫ ( ) called the indefinite integral, to represents the family of
all anti-derivatives of ( ), and we write
Mathematical Analysis 56
, ( )
( ) = ( )+ = ( ).
Here ∫ is called the integral sign and indicates that anti-derivative is performed
with respect to and is called the integration constant. We read this as “the integral
of ( ) with respect to ” or “the integral of ( ) ”. In other words ∫ ( )
means the general anti-derivative of ( ) including integration constant.
Some properties of indefinite integral
a)
( ) = ( ) = ( )
b)
( ) = ( )
c)
( )± ( ) = ( ) ± ( )
Indefinite Integral of Basic Functions
( )
( )
+ , ≠ −1
+1
1 2√ +
√
1 | |+ , ≠0
+ , >0
+
sin − cos +
cos sin +
tan ln| |+
csc ln|csc − cot | +
Mathematical Analysis 57
, sec ln|sec + tan | +
cot ln|sin | +
sec tan +
csc − cot +
sec tan sec +
csc cot − csc +
sinh cosh +
cosh sinh +
tanh ln|cosh | +
csch ln tanh +
2
sech tan (sinh ) +
coth ln|sinh | +
sech tanh +
csch − coth +
sech tanh − sech +
csch coth − csch +
1 sin +
√1 − or
− cos +
⎧ sin +
1
or
√ − ⎨
⎩− cos +
1 tan +
1+ or
− cot +
1
⎧ tan +
1 ⎪
+ or
⎨ 1
⎪− cot +
⎩
1 sec +
√ −1 or
− csc +
Mathematical Analysis 58
Integration
5.1 Anti-Derivatives
Many operations in mathematics have reverses – compare addition and subtraction,
multiplication and division, and powers and roots. We now know how to find the
derivatives of many functions. The reverse operation, anti-derivative will receive our
attention in this chapter. A function is an anti-derivative (or indefinite integral) of
if
= ,
we call the anti-derivative (or indefinite integral) of . For example, if ( ) = , we
can find its anti-derivative by realizing that for the function ( ) =
1 1
= = .2 = = ( ).
2 2
Thus ( ) = is an anti-derivative of ( ) = .
However, if is a constant:
1 1
+ = .2 = .
2 2
Since the derivative of a constant is zero. The family of all anti-derivatives of is thus
+ , where can be any constant.
Note that you should always check an anti-derivative by differentiating it and seeing
that you recover .
5.2 Indefinite integral
We use the notation ∫ ( ) called the indefinite integral, to represents the family of
all anti-derivatives of ( ), and we write
Mathematical Analysis 56
, ( )
( ) = ( )+ = ( ).
Here ∫ is called the integral sign and indicates that anti-derivative is performed
with respect to and is called the integration constant. We read this as “the integral
of ( ) with respect to ” or “the integral of ( ) ”. In other words ∫ ( )
means the general anti-derivative of ( ) including integration constant.
Some properties of indefinite integral
a)
( ) = ( ) = ( )
b)
( ) = ( )
c)
( )± ( ) = ( ) ± ( )
Indefinite Integral of Basic Functions
( )
( )
+ , ≠ −1
+1
1 2√ +
√
1 | |+ , ≠0
+ , >0
+
sin − cos +
cos sin +
tan ln| |+
csc ln|csc − cot | +
Mathematical Analysis 57
, sec ln|sec + tan | +
cot ln|sin | +
sec tan +
csc − cot +
sec tan sec +
csc cot − csc +
sinh cosh +
cosh sinh +
tanh ln|cosh | +
csch ln tanh +
2
sech tan (sinh ) +
coth ln|sinh | +
sech tanh +
csch − coth +
sech tanh − sech +
csch coth − csch +
1 sin +
√1 − or
− cos +
⎧ sin +
1
or
√ − ⎨
⎩− cos +
1 tan +
1+ or
− cot +
1
⎧ tan +
1 ⎪
+ or
⎨ 1
⎪− cot +
⎩
1 sec +
√ −1 or
− csc +
Mathematical Analysis 58
