INTEGRALS
First fundamental theorem of integral calculus
Let the area function be defined by A(x) = for all x ≥ a, where
The function f is assumed to be continuous on [a, b]. then A’(x) = f(x)for all x ∈ [a, b].
Second fundamental theorem of integral calculus
Let f be a continuous function of x defined on the closed interval [a, b] and let F be
another
function
such
that
f,
then
This is called the definite integral of f over the range [a, b] where a and b are called the
limits of integration, a being the lower limit and b the upper limit.
The function under the sign of integration is called integrand.
Element of Integration::
In the integral f(x) d x, dx is known as element of integration and it indicates the variable
with respect to which the given function is to be integrated. In the integral cos θ d θ, d θ
indicates that cos θ is to be integrated w.r.t.
w.r.t.θ.
Derivative, Primitive and Integration:
We have read in differential calculus that, if
, then is known as the
derivative of f(x) and f(x) is called a primitive or an integral of F(x). The process of finding
the primitive is called Integration.
Selection for Proper Substitution:
There is no hard and fast rule for making suitable substitutions. However on the basis of
experience, some useful suggestions are given below:
♦
If the integrand contains t-rations of f(x) or logarithm of f(x) or an exponential
function in which the index is f(x), put f(x) = t.
♦
If the integrand is a rations function of ex, put ex = t.
♦
To evaluate:
sinnx dx, where n is positive odd integer, put cos x = t.
cosnx dx, where n is positive odd integer, put cos x =t.
secnx dx, where n is positive even integer, put tan x = t.
cosecnx dx, where n is positive even integer, put cot x = t.
sinpx cosq dx, where q is positive odd integer, put sin x = t.
sinpx cosq dx, where p is positive odd integer, put sin x = t.
Derivatives
(i)
Integrals (Anti derivatives)
d x &'(
%
* x & ;
dx n 1
x &'(
x dx C, n ≠ 1
n1
d
x 1;
dx
dx x C
Particularly, we note that
&
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
d
sin x cos x ;
dx
cos xdx sin x C
d
tan x sec 3 x;
dx
sec 3 xdx tan x C
d
cos x sin x ;
dx
d
cot x cosec 3 x;
dx
d
sec x sec x tan x;
dx
d
cosecx cosecx cot x;
dx
d
1
sin5( x dx
√1 x 3
d
1
cos 5( x ;
dx
√1 x 3
d
1
tan5( x ;
dx
1 x3
d
1
cot 5( x ;
dx
1 x3
d
1
sec 5( x ;
dx
x √x 35(
d
1
cosec 5( x ;
dx
x √x 35(
d 7
e e7 ;
dx
d
1
log |x| ;
dx
x
d a7
;
< a7 ;
dx log a
sin xdx cos x C
cosec 3 xdx cot x C
4 sec x tan xdx sec x C
4 cosecx cot xdx cosecx C
dx
√1 1
√1 x3
x3
cos 5( x C
cos 5( x C
dx
tan5( x C
1 x3
dx
cot 5( x C
3
1x
dx
√x 35(
x
dx
√x 35(
x
sec 5( x C
cosec 5( x C
e7 dx e7 C
4
1
dx log|x| C
x
a7
a dx C
log a
7
Some standard integrals
(i)
= >?@
='(
, ≠ 1. BCDE. (ii)
cos sin (iii)
(iv)
sec 3 tan (v)
(vi)
sec tan sec (vii)
(ix)
(xi)
(xiii)
(xv)
4 CFC cot 4
4
√1 3
CFC CF 5( C 5( 1 3
4 log 3
4
(viii)
√ 3 1
CFC 5( (x)
(xii)
(xiv)
(xvi)
4 sin cos 4 CFC 3 C 4
4
√1 3
F5( 5( 1 3
4 4
4
√ 3 1
FC 5( 1
G. || Integrals of some special functions
H 5H 3 log I'I (
5
H 5 H 3 log I5I (
'
H 'H 5( (
√ H 5H logJ √ 3 3 J √H 5 H F5( √ H 'H logJ √ 3 3 J Definite Integrals:
Let ∅ be the primitive of a function f(x) defined on [a, b] i.e., L∅M f(x). Then the
definite integral of f(x) over [a, b] is denoted by and is defined as ∅ ∅
i.e.,
♦
=∅ ∅, where ∅
0
Properties of Definite Integrals:
♦
♦
♦
♦
♦
♦
♦
♦
♦
= 0
= = C
+ = C
= 0
0
= C
2
= + 2 0
0
0
2
2 , 2 T
= R 0
0
0, if2 2 , ifisanevenfunctionT
= R
0
0, ifisanoddfunction
The area of the region bounded by y2 = 4ax and y = m x is
WXY
Z[Z
square units.
The area of the region bounded by y2 = 4 ax and its latus rectum is
WXY
Z
square units.
Y
The area of the region bounded by one are of sin (a x) or cos (a x) and x-axis is X sq. units.
Area of the ellipse
Y
\Y
X
]Y
^Y
_`abX^cd. efghc.
\i j\ k\ j\ \i i. \i5_ ii _\i5Y ⋯ k\
\Y \i '_
k\
5i i
n
im_ _ \
_
\Y _'\i i
_
i
_5i . _ \5i _5i n
_
_
p j5X\ cgf ^\k\ Y Y X > p
X '^
^Y
∞
Xj\'^jm\
qj\ 'kjm\
Xj\'^jm\
qj\ 'kjm\
k\
k\
=Y rs q kt \ s q kt uvw|qj\ kj5\ | nT
_
X
^
X
^
i
Area of the region bounded by y = ax – [x]; x = 0, x = n and y = 0 is uvw X
x\'y
zX\Y '^\'q
k\ X √X\Y ^\ q sy YXt {_ where I1 = cosh-1s
x
x^
0
k\
zX\Y ^\'q
_
z|X|
X vc \'^a`i\
q vc \'k cgf \
a`i5_ }
~ n
YX\'^
√∆
k\ = snY'XY t \ sqY'XYt uvw|q vc \ k cgf \| n
Xq'^k
Xk5q
Area bounded by the curve y2 = 4ax and the line y = mx is Z[Z .
p
b
k\
X5vc \
b
zXY 5_
WXY
t,if a > 0, ∆ >
YX\'^
√∆
p j5\ \i k\ i_ j5X ;_ X YT ZT … iT <T
X
XY
XZ
Xi
p \i k\,
jjX >
X
Symbols/Terms/Phrases
Meaning
\`i \k\
Integral of f with respect to x
\k\
\`i \k\
Integrand
Integrate
Find the integral
An integral of f
A function F such that F’(x) = f(x)
Integration
The
Variable of integration
process
of
finding
the
integral
Constant of Integration
Any real number C, considered as
constant function
S.No.
1.
2.
3.
4.
5.
Form of the rational function
, ≠ Form of the partial fraction
3
3 C
3
C
3 3 3 3 C
Where x2 + bx + c cannot be
factorized further
3
3
C
The method of resolving rational functions into partial factors:
, ≠ Type of proper rational function
H ''
, , , Care
555
H ''
, , ≠
5H 5
3
Types of partial fractions
distinct
where 3 C cannot be
C
3
C
factorised
C
H ' H ''
3
3
3
3
, where , C H '' H ''
H ''
,
5 H ''
C cannot be factorised
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