Detailed Notes: Indefinite Integration | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


INTRODUCTION 
Indefinite integration is a fundamental concept in calculus that deals with finding 
antiderivatives of functions. It's essentially the reverse process of differentiation, 
allowing us to work backwards from a given function to find the original function it came 
from. At its core, indefinite integration seeks to answer the question: "What function, 
when differentiated, gives us this particular function?" 
The power of indefinite integration lies in its ability to solve complex problems involving 
rates of change, accumulation, and area calculations. It provides a set of tools and 
techniques that enable us to tackle a variety of mathematical challenges, from simple 
polynomial functions to more complex trigonometric and exponential expressions. This 
process is crucial in mathematics and has wide-ranging applications in physics, 
engineering, and other sciences. 
Understanding indefinite integration opens up a world of mathematical possibilities. It's 
a key stepping stone to more advanced calculus concepts and forms the basis for solving 
differential equations, which are ubiquitous in describing real-world phenomena. As 
students delve into this topic, they'll discover a blend of logic, pattern recognition, and 
creative problem-solving that makes indefinite integration both challenging and 
rewarding. 
INDEFINITE INTEGRATION 
If ?? & ?? are function of ?? such that ?? '
( ?? ) = ?? ( ?? ) then the function ?? is called a 
PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of ?? ( ?? ) w.r.t. ?? and is written 
symbolically as ??? ( ?? ) ???? = ?? ( ?? )+ ?? ?
?? ????
{?? ( ?? )+ ?? } = ?? ( ?? ) , where ?? is called the 
constant of integration. 
1. GEOMETRICAL INTERPRETATION 
OF INDEFINITE INTEGRAL: 
??? ( ?? ) ???? = ?? ( ?? )+ ?? = ?? ( say) , represents a family of curves. The different values of ?? 
will correspond to different members of this family and these members can be obtained 
by shifting any one of the curves parallel to itself. This is the geometrical interpretation 
of indefinite integral. 
Let ?? ( ?? ) = 2?? . Then ??? ( ?? ) ???? = ?? 2
+ ?? . For different values of ?? , we get different 
integrals. But these integrals are very similar geometrically. 
Thus, ?? = ?? 2
+ ?? , where ?? is arbitrary constant, represents a family of integrals. By 
assigning different values to ?? , we get different members of the family. These together 
constitute the indefinite integral. In this case, each integral represents a parabola with its 
axis along y-axis. 
Page 2


INTRODUCTION 
Indefinite integration is a fundamental concept in calculus that deals with finding 
antiderivatives of functions. It's essentially the reverse process of differentiation, 
allowing us to work backwards from a given function to find the original function it came 
from. At its core, indefinite integration seeks to answer the question: "What function, 
when differentiated, gives us this particular function?" 
The power of indefinite integration lies in its ability to solve complex problems involving 
rates of change, accumulation, and area calculations. It provides a set of tools and 
techniques that enable us to tackle a variety of mathematical challenges, from simple 
polynomial functions to more complex trigonometric and exponential expressions. This 
process is crucial in mathematics and has wide-ranging applications in physics, 
engineering, and other sciences. 
Understanding indefinite integration opens up a world of mathematical possibilities. It's 
a key stepping stone to more advanced calculus concepts and forms the basis for solving 
differential equations, which are ubiquitous in describing real-world phenomena. As 
students delve into this topic, they'll discover a blend of logic, pattern recognition, and 
creative problem-solving that makes indefinite integration both challenging and 
rewarding. 
INDEFINITE INTEGRATION 
If ?? & ?? are function of ?? such that ?? '
( ?? ) = ?? ( ?? ) then the function ?? is called a 
PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of ?? ( ?? ) w.r.t. ?? and is written 
symbolically as ??? ( ?? ) ???? = ?? ( ?? )+ ?? ?
?? ????
{?? ( ?? )+ ?? } = ?? ( ?? ) , where ?? is called the 
constant of integration. 
1. GEOMETRICAL INTERPRETATION 
OF INDEFINITE INTEGRAL: 
??? ( ?? ) ???? = ?? ( ?? )+ ?? = ?? ( say) , represents a family of curves. The different values of ?? 
will correspond to different members of this family and these members can be obtained 
by shifting any one of the curves parallel to itself. This is the geometrical interpretation 
of indefinite integral. 
Let ?? ( ?? ) = 2?? . Then ??? ( ?? ) ???? = ?? 2
+ ?? . For different values of ?? , we get different 
integrals. But these integrals are very similar geometrically. 
Thus, ?? = ?? 2
+ ?? , where ?? is arbitrary constant, represents a family of integrals. By 
assigning different values to ?? , we get different members of the family. These together 
constitute the indefinite integral. In this case, each integral represents a parabola with its 
axis along y-axis. 
If the line ?? = ?? intersects the parabolas ?? = ?? 2
, ?? = ?? 2
+ 1, ?? = ?? 2
+ 2, ?? = ?? 2
- 1, ?? =
?? 2
- 2 at ?? 0
, ?? 1
, ?? 2
, ?? -1
, ?? -2
 etc., respectively, then 
????
????
 at these points equals 2?? . This 
indicates that the tangents to the curves at these points are parallel. Thus, ?2?????? = ?? 2
+
?? = ?? ( ?? )+ ?? (say), implies that the tangents to all the curves ?? ( ?? )+ ?? , ?? ? ?? , at the 
points of intersection of the curves by the line 
 
 
?? = ?? , ( ?? ? ?? ) , are parallel. 
2. STANDARD FORMULAE: 
(i)  ?( ???? + ?? )
?? ???? =
( ???? +?? )
?? +1
?? ( ?? +1)
+ ?? ; ?? ? -1 
 
Page 3


INTRODUCTION 
Indefinite integration is a fundamental concept in calculus that deals with finding 
antiderivatives of functions. It's essentially the reverse process of differentiation, 
allowing us to work backwards from a given function to find the original function it came 
from. At its core, indefinite integration seeks to answer the question: "What function, 
when differentiated, gives us this particular function?" 
The power of indefinite integration lies in its ability to solve complex problems involving 
rates of change, accumulation, and area calculations. It provides a set of tools and 
techniques that enable us to tackle a variety of mathematical challenges, from simple 
polynomial functions to more complex trigonometric and exponential expressions. This 
process is crucial in mathematics and has wide-ranging applications in physics, 
engineering, and other sciences. 
Understanding indefinite integration opens up a world of mathematical possibilities. It's 
a key stepping stone to more advanced calculus concepts and forms the basis for solving 
differential equations, which are ubiquitous in describing real-world phenomena. As 
students delve into this topic, they'll discover a blend of logic, pattern recognition, and 
creative problem-solving that makes indefinite integration both challenging and 
rewarding. 
INDEFINITE INTEGRATION 
If ?? & ?? are function of ?? such that ?? '
( ?? ) = ?? ( ?? ) then the function ?? is called a 
PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of ?? ( ?? ) w.r.t. ?? and is written 
symbolically as ??? ( ?? ) ???? = ?? ( ?? )+ ?? ?
?? ????
{?? ( ?? )+ ?? } = ?? ( ?? ) , where ?? is called the 
constant of integration. 
1. GEOMETRICAL INTERPRETATION 
OF INDEFINITE INTEGRAL: 
??? ( ?? ) ???? = ?? ( ?? )+ ?? = ?? ( say) , represents a family of curves. The different values of ?? 
will correspond to different members of this family and these members can be obtained 
by shifting any one of the curves parallel to itself. This is the geometrical interpretation 
of indefinite integral. 
Let ?? ( ?? ) = 2?? . Then ??? ( ?? ) ???? = ?? 2
+ ?? . For different values of ?? , we get different 
integrals. But these integrals are very similar geometrically. 
Thus, ?? = ?? 2
+ ?? , where ?? is arbitrary constant, represents a family of integrals. By 
assigning different values to ?? , we get different members of the family. These together 
constitute the indefinite integral. In this case, each integral represents a parabola with its 
axis along y-axis. 
If the line ?? = ?? intersects the parabolas ?? = ?? 2
, ?? = ?? 2
+ 1, ?? = ?? 2
+ 2, ?? = ?? 2
- 1, ?? =
?? 2
- 2 at ?? 0
, ?? 1
, ?? 2
, ?? -1
, ?? -2
 etc., respectively, then 
????
????
 at these points equals 2?? . This 
indicates that the tangents to the curves at these points are parallel. Thus, ?2?????? = ?? 2
+
?? = ?? ( ?? )+ ?? (say), implies that the tangents to all the curves ?? ( ?? )+ ?? , ?? ? ?? , at the 
points of intersection of the curves by the line 
 
 
?? = ?? , ( ?? ? ?? ) , are parallel. 
2. STANDARD FORMULAE: 
(i)  ?( ???? + ?? )
?? ???? =
( ???? +?? )
?? +1
?? ( ?? +1)
+ ?? ; ?? ? -1 
 
(iii) ??? ???? +?? ???? =
1
?? ?? ???? +?? + ?? 
(iv) ??? ???? +?? ???? =
1
?? ?? ???? +?? ????
+ ?? , ( ?? > 0) 
(v) ??????? ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(vi) ??????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(vii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(viii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(ix) ??????? 2
 ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(x) ??????????? 2
 ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(xi) ??????????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? = -
1
?? ?????????? ( ???? + ?? )+ ?? 
(xii) ??????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(xiii) ??????? ?????? = ???? |?????? ?? + ?????? ?? | + ?? = ???? |?????? (
?? 4
+
?? 2
)| + ?? 
(xiv) ??????????? ?????? = ?????? |?????????? ?? - ?????? ?? | + ?? = ???? |?????? 
?? 2
| + ?? = -???? |?????????? ?? + ?????? ?? | +
?? 
(xv) ?
????
v?? 2
-?? 2
= ?????? -1
 
?? ?? + ?? 
(xvi) ?
????
?? 2
+?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xvii) ?
????
?? v?? 2
-?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xviii) ?
????
v?? 2
+?? 2
= ???? [?? + v ?? 2
+ ?? 2
] + ?? 
(xix) ?
????
v?? 2
-?? 2
= ???? [?? + v ?? 2
- ?? 2
] + ?? 
(xx) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? +?? ?? -?? | + ?? 
(xxi) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? -?? ?? +?? | + ?? 
(xxii) ?v ?? 2
- ?? 2
???? =
?? 2
v ?? 2
- ?? 2
+
?? 2
2
?????? -1
 
?? ?? + ?? 
(xxiii) ?v ?? 2
+ ?? 2
???? =
?? 2
v ?? 2
+ ?? 2
+
?? 2
2
???? ( ?? + v ?? 2
+ ?? 2
)+ ?? 
Page 4


INTRODUCTION 
Indefinite integration is a fundamental concept in calculus that deals with finding 
antiderivatives of functions. It's essentially the reverse process of differentiation, 
allowing us to work backwards from a given function to find the original function it came 
from. At its core, indefinite integration seeks to answer the question: "What function, 
when differentiated, gives us this particular function?" 
The power of indefinite integration lies in its ability to solve complex problems involving 
rates of change, accumulation, and area calculations. It provides a set of tools and 
techniques that enable us to tackle a variety of mathematical challenges, from simple 
polynomial functions to more complex trigonometric and exponential expressions. This 
process is crucial in mathematics and has wide-ranging applications in physics, 
engineering, and other sciences. 
Understanding indefinite integration opens up a world of mathematical possibilities. It's 
a key stepping stone to more advanced calculus concepts and forms the basis for solving 
differential equations, which are ubiquitous in describing real-world phenomena. As 
students delve into this topic, they'll discover a blend of logic, pattern recognition, and 
creative problem-solving that makes indefinite integration both challenging and 
rewarding. 
INDEFINITE INTEGRATION 
If ?? & ?? are function of ?? such that ?? '
( ?? ) = ?? ( ?? ) then the function ?? is called a 
PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of ?? ( ?? ) w.r.t. ?? and is written 
symbolically as ??? ( ?? ) ???? = ?? ( ?? )+ ?? ?
?? ????
{?? ( ?? )+ ?? } = ?? ( ?? ) , where ?? is called the 
constant of integration. 
1. GEOMETRICAL INTERPRETATION 
OF INDEFINITE INTEGRAL: 
??? ( ?? ) ???? = ?? ( ?? )+ ?? = ?? ( say) , represents a family of curves. The different values of ?? 
will correspond to different members of this family and these members can be obtained 
by shifting any one of the curves parallel to itself. This is the geometrical interpretation 
of indefinite integral. 
Let ?? ( ?? ) = 2?? . Then ??? ( ?? ) ???? = ?? 2
+ ?? . For different values of ?? , we get different 
integrals. But these integrals are very similar geometrically. 
Thus, ?? = ?? 2
+ ?? , where ?? is arbitrary constant, represents a family of integrals. By 
assigning different values to ?? , we get different members of the family. These together 
constitute the indefinite integral. In this case, each integral represents a parabola with its 
axis along y-axis. 
If the line ?? = ?? intersects the parabolas ?? = ?? 2
, ?? = ?? 2
+ 1, ?? = ?? 2
+ 2, ?? = ?? 2
- 1, ?? =
?? 2
- 2 at ?? 0
, ?? 1
, ?? 2
, ?? -1
, ?? -2
 etc., respectively, then 
????
????
 at these points equals 2?? . This 
indicates that the tangents to the curves at these points are parallel. Thus, ?2?????? = ?? 2
+
?? = ?? ( ?? )+ ?? (say), implies that the tangents to all the curves ?? ( ?? )+ ?? , ?? ? ?? , at the 
points of intersection of the curves by the line 
 
 
?? = ?? , ( ?? ? ?? ) , are parallel. 
2. STANDARD FORMULAE: 
(i)  ?( ???? + ?? )
?? ???? =
( ???? +?? )
?? +1
?? ( ?? +1)
+ ?? ; ?? ? -1 
 
(iii) ??? ???? +?? ???? =
1
?? ?? ???? +?? + ?? 
(iv) ??? ???? +?? ???? =
1
?? ?? ???? +?? ????
+ ?? , ( ?? > 0) 
(v) ??????? ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(vi) ??????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(vii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(viii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(ix) ??????? 2
 ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(x) ??????????? 2
 ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(xi) ??????????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? = -
1
?? ?????????? ( ???? + ?? )+ ?? 
(xii) ??????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(xiii) ??????? ?????? = ???? |?????? ?? + ?????? ?? | + ?? = ???? |?????? (
?? 4
+
?? 2
)| + ?? 
(xiv) ??????????? ?????? = ?????? |?????????? ?? - ?????? ?? | + ?? = ???? |?????? 
?? 2
| + ?? = -???? |?????????? ?? + ?????? ?? | +
?? 
(xv) ?
????
v?? 2
-?? 2
= ?????? -1
 
?? ?? + ?? 
(xvi) ?
????
?? 2
+?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xvii) ?
????
?? v?? 2
-?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xviii) ?
????
v?? 2
+?? 2
= ???? [?? + v ?? 2
+ ?? 2
] + ?? 
(xix) ?
????
v?? 2
-?? 2
= ???? [?? + v ?? 2
- ?? 2
] + ?? 
(xx) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? +?? ?? -?? | + ?? 
(xxi) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? -?? ?? +?? | + ?? 
(xxii) ?v ?? 2
- ?? 2
???? =
?? 2
v ?? 2
- ?? 2
+
?? 2
2
?????? -1
 
?? ?? + ?? 
(xxiii) ?v ?? 2
+ ?? 2
???? =
?? 2
v ?? 2
+ ?? 2
+
?? 2
2
???? ( ?? + v ?? 2
+ ?? 2
)+ ?? 
(xxiv) ?v ?? 2
- ?? 2
???? =
?? 2
v ?? 2
- ?? 2
-
?? 2
2
???? ( ?? + v ?? 2
- ?? 2
)+ ?? 
( ?????? ) ??? ????
· ?????? ???????? =
?? ????
?? 2
+ ?? 2
( ???????? ???? - ???????? ???? )+ ?? =
?? ????
v ?? 2
+ ?? 2
?????? (???? - ?????? -1
 
?? ?? ) + ?? 
(xxvi) ??? ????
· ?????? ???????? =
?? ????
?? 2
+?? 2
( ???????? ???? + ???????? ???? )+ ?? =
?? ????
v?? 2
+?? 2
?????? (???? - ?????? -1
 
?? ?? ) + ?? 
3. TECHNIQUES OF INTEGRATION: 
(a) Substitution or change of independent variable: 
If ?? ( ?? ) is a continuous differentiable function, then to evaluate integrals of the form 
??? ( ?? ( ?? ) ) ?? '
( ?? ) ???? , we substitute ?? ( ?? )= ?? and ?? '
( ?? ) ???? = ???? . 
Hence ?? = ??? ( ?? ( ?? ) ) ?? '
( ?? ) ???? reduces to ??? ( ?? ) ???? . 
(i) Fundamental deductions of method of substitution: 
?[?? ( ?? ) ]
?? ?? '
( ?? ) ???? OR ?
?? '
( ?? )
[?? ( ?? ) ]
?? ???? put ?? ( ?? )= ?? & proceed. 
Problem 1: Evaluate ?
?????? 3
 ?? ?????? 2
 ?? +?????? ?? ???? 
Solution:  ?? = ?
( 1-?????? 2
 ?? ) ?????? ?? ?????? ?? ( 1+?????? ?? )
???? = ?
1-?????? ?? ?????? ?? ?????? ?????? 
Put ?????? ?? = ?? ? ?????? ?????? = ???? 
? ?? = ?
1 - ?? ?? ???? = ???? |?? | - ?? + ?? = ???? |?????? ?? | - ?????? ?? + ?? 
Ans. 
Problem 2: Evaluate ?
( ?? 2
-1) ????
( ?? 4
+3?? 2
+1) ?????? -1
 (?? +
1
?? )
 
Solution: The given integral can be written as 
?? = ?
(1 -
1
?? 2
)????
[(?? +
1
?? )
2
+ 1] ?????? -1
 (?? +
1
?? )
 
Let (?? +
1
?? ) = ?? . Differentiating we get (1 -
1
?? 2
)???? = ???? 
Hence ?? = ?
????
( ?? 2
+1) ?????? -1
 ?? 
Page 5


INTRODUCTION 
Indefinite integration is a fundamental concept in calculus that deals with finding 
antiderivatives of functions. It's essentially the reverse process of differentiation, 
allowing us to work backwards from a given function to find the original function it came 
from. At its core, indefinite integration seeks to answer the question: "What function, 
when differentiated, gives us this particular function?" 
The power of indefinite integration lies in its ability to solve complex problems involving 
rates of change, accumulation, and area calculations. It provides a set of tools and 
techniques that enable us to tackle a variety of mathematical challenges, from simple 
polynomial functions to more complex trigonometric and exponential expressions. This 
process is crucial in mathematics and has wide-ranging applications in physics, 
engineering, and other sciences. 
Understanding indefinite integration opens up a world of mathematical possibilities. It's 
a key stepping stone to more advanced calculus concepts and forms the basis for solving 
differential equations, which are ubiquitous in describing real-world phenomena. As 
students delve into this topic, they'll discover a blend of logic, pattern recognition, and 
creative problem-solving that makes indefinite integration both challenging and 
rewarding. 
INDEFINITE INTEGRATION 
If ?? & ?? are function of ?? such that ?? '
( ?? ) = ?? ( ?? ) then the function ?? is called a 
PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of ?? ( ?? ) w.r.t. ?? and is written 
symbolically as ??? ( ?? ) ???? = ?? ( ?? )+ ?? ?
?? ????
{?? ( ?? )+ ?? } = ?? ( ?? ) , where ?? is called the 
constant of integration. 
1. GEOMETRICAL INTERPRETATION 
OF INDEFINITE INTEGRAL: 
??? ( ?? ) ???? = ?? ( ?? )+ ?? = ?? ( say) , represents a family of curves. The different values of ?? 
will correspond to different members of this family and these members can be obtained 
by shifting any one of the curves parallel to itself. This is the geometrical interpretation 
of indefinite integral. 
Let ?? ( ?? ) = 2?? . Then ??? ( ?? ) ???? = ?? 2
+ ?? . For different values of ?? , we get different 
integrals. But these integrals are very similar geometrically. 
Thus, ?? = ?? 2
+ ?? , where ?? is arbitrary constant, represents a family of integrals. By 
assigning different values to ?? , we get different members of the family. These together 
constitute the indefinite integral. In this case, each integral represents a parabola with its 
axis along y-axis. 
If the line ?? = ?? intersects the parabolas ?? = ?? 2
, ?? = ?? 2
+ 1, ?? = ?? 2
+ 2, ?? = ?? 2
- 1, ?? =
?? 2
- 2 at ?? 0
, ?? 1
, ?? 2
, ?? -1
, ?? -2
 etc., respectively, then 
????
????
 at these points equals 2?? . This 
indicates that the tangents to the curves at these points are parallel. Thus, ?2?????? = ?? 2
+
?? = ?? ( ?? )+ ?? (say), implies that the tangents to all the curves ?? ( ?? )+ ?? , ?? ? ?? , at the 
points of intersection of the curves by the line 
 
 
?? = ?? , ( ?? ? ?? ) , are parallel. 
2. STANDARD FORMULAE: 
(i)  ?( ???? + ?? )
?? ???? =
( ???? +?? )
?? +1
?? ( ?? +1)
+ ?? ; ?? ? -1 
 
(iii) ??? ???? +?? ???? =
1
?? ?? ???? +?? + ?? 
(iv) ??? ???? +?? ???? =
1
?? ?? ???? +?? ????
+ ?? , ( ?? > 0) 
(v) ??????? ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(vi) ??????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(vii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(viii) ??????? ( ???? + ?? ) ???? =
1
?? ???? |?????? ( ???? + ?? ) | + ?? 
(ix) ??????? 2
 ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(x) ??????????? 2
 ( ???? + ?? ) ???? = -
1
?? ?????? ( ???? + ?? )+ ?? 
(xi) ??????????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? = -
1
?? ?????????? ( ???? + ?? )+ ?? 
(xii) ??????? ( ???? + ?? )· ?????? ( ???? + ?? ) ???? =
1
?? ?????? ( ???? + ?? )+ ?? 
(xiii) ??????? ?????? = ???? |?????? ?? + ?????? ?? | + ?? = ???? |?????? (
?? 4
+
?? 2
)| + ?? 
(xiv) ??????????? ?????? = ?????? |?????????? ?? - ?????? ?? | + ?? = ???? |?????? 
?? 2
| + ?? = -???? |?????????? ?? + ?????? ?? | +
?? 
(xv) ?
????
v?? 2
-?? 2
= ?????? -1
 
?? ?? + ?? 
(xvi) ?
????
?? 2
+?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xvii) ?
????
?? v?? 2
-?? 2
=
1
?? ?????? -1
 
?? ?? + ?? 
(xviii) ?
????
v?? 2
+?? 2
= ???? [?? + v ?? 2
+ ?? 2
] + ?? 
(xix) ?
????
v?? 2
-?? 2
= ???? [?? + v ?? 2
- ?? 2
] + ?? 
(xx) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? +?? ?? -?? | + ?? 
(xxi) ?
????
?? 2
-?? 2
=
1
2?? ???? |
?? -?? ?? +?? | + ?? 
(xxii) ?v ?? 2
- ?? 2
???? =
?? 2
v ?? 2
- ?? 2
+
?? 2
2
?????? -1
 
?? ?? + ?? 
(xxiii) ?v ?? 2
+ ?? 2
???? =
?? 2
v ?? 2
+ ?? 2
+
?? 2
2
???? ( ?? + v ?? 2
+ ?? 2
)+ ?? 
(xxiv) ?v ?? 2
- ?? 2
???? =
?? 2
v ?? 2
- ?? 2
-
?? 2
2
???? ( ?? + v ?? 2
- ?? 2
)+ ?? 
( ?????? ) ??? ????
· ?????? ???????? =
?? ????
?? 2
+ ?? 2
( ???????? ???? - ???????? ???? )+ ?? =
?? ????
v ?? 2
+ ?? 2
?????? (???? - ?????? -1
 
?? ?? ) + ?? 
(xxvi) ??? ????
· ?????? ???????? =
?? ????
?? 2
+?? 2
( ???????? ???? + ???????? ???? )+ ?? =
?? ????
v?? 2
+?? 2
?????? (???? - ?????? -1
 
?? ?? ) + ?? 
3. TECHNIQUES OF INTEGRATION: 
(a) Substitution or change of independent variable: 
If ?? ( ?? ) is a continuous differentiable function, then to evaluate integrals of the form 
??? ( ?? ( ?? ) ) ?? '
( ?? ) ???? , we substitute ?? ( ?? )= ?? and ?? '
( ?? ) ???? = ???? . 
Hence ?? = ??? ( ?? ( ?? ) ) ?? '
( ?? ) ???? reduces to ??? ( ?? ) ???? . 
(i) Fundamental deductions of method of substitution: 
?[?? ( ?? ) ]
?? ?? '
( ?? ) ???? OR ?
?? '
( ?? )
[?? ( ?? ) ]
?? ???? put ?? ( ?? )= ?? & proceed. 
Problem 1: Evaluate ?
?????? 3
 ?? ?????? 2
 ?? +?????? ?? ???? 
Solution:  ?? = ?
( 1-?????? 2
 ?? ) ?????? ?? ?????? ?? ( 1+?????? ?? )
???? = ?
1-?????? ?? ?????? ?? ?????? ?????? 
Put ?????? ?? = ?? ? ?????? ?????? = ???? 
? ?? = ?
1 - ?? ?? ???? = ???? |?? | - ?? + ?? = ???? |?????? ?? | - ?????? ?? + ?? 
Ans. 
Problem 2: Evaluate ?
( ?? 2
-1) ????
( ?? 4
+3?? 2
+1) ?????? -1
 (?? +
1
?? )
 
Solution: The given integral can be written as 
?? = ?
(1 -
1
?? 2
)????
[(?? +
1
?? )
2
+ 1] ?????? -1
 (?? +
1
?? )
 
Let (?? +
1
?? ) = ?? . Differentiating we get (1 -
1
?? 2
)???? = ???? 
Hence ?? = ?
????
( ?? 2
+1) ?????? -1
 ?? 
Now make one more substitution ?????? -1
 ?? = ?? . Then 
????
?? 2
+1
= ???? and ?? = ?
????
?? = ???? |?? | + ?? 
Returning to ?? , and then to ?? , we have 
?? = ???? |?????? -1
 ?? | + ?? = ???? |?????? -1
 (?? +
1
?? )| + ?? #( ?????? . )  
Problem 4: Evaluate ?v
1-v ?? 1+v ?? ·
1
?? ???? 
Solution:   Put ?? = ?????? 2
 ?? ? ???? = -2?????? ???????? ?????? 
? ??  = ? ?
?
?
 v
1 - ?????? ?? 1 + ?????? ?? ·
1
?????? 2
 ?? ( -2?????? ???????? ?? ) ???? = - ? ?
?
?
 2?????? 
?? 2
?????? ??????   
= -4 ? ?
?
?
 
?????? 2
 ( ?? /2)
?????? ?? ???? = -2 ? ?
?
?
 
1 - ?????? ?? ?????? ?? ????
= -2???? |?????? ?? + ?????? ?? | + 2?? + ??   = -2???? |
1 + v 1 - ?? v ?? | + 2?????? -1
 v ?? + ??  
Do yourself -2: 
(i) Evaluate:?v
?? -3
2-?? ???? 
(ii) Evaluate: ?
????
?? v?? 2
+4
 
(b) Integration by part: ??? . ?????? = ?? ??????? - ? [
????
????
· ??????? ] ???? where ?? &?? are 
differentiable functions and are commonly designated as first & second function 
respectively. 
Note: While using integration by parts, choose ?? &?? such that 
 ( ?? ) ? ?
?
?
 ?????? &  ( ???? ) ? ?
?
?
 [
????
????
? ?
?
?
 ?????? ] ???? ?????? ???????????? ???? ?????????????????? .   
This is generally obtained by choosing first function as the function which comes first in 
the word ILATE, where; I-Inverse function, L-Logarithmic function, A-Algebraic 
function, T-Trigonometric function & E-Exponential function. 
Problem 5: Evaluate: ??????? v ?? ???? 
Solution:   Consider  ?? = ??????? v ?? ???? 
Let  v ?? = ??  then 
1
2v ?? ???? = ???? 
i.e.  ???? = 2v ?? ???? 
or ???? = 2?????? 
so