How to Get Over indefinite Integration ?
In this article , we will cover the topic Indefinite Integration. This has been written specially taking care of needs & demands of Engineering Aspirants and everything necessary has been added . In the next article i will discuss some interesting problems on Indefinite Integration . In the previous article i have already discussed Differential Calculus .
Primitive or Anti-derivative.
The integral or primitive or anti-derivative of a function f (x) w.r.t. x is the function 
whose derivative w.r.t. x is f(x) and we write it as 
.
The process of finding the function 
is called “integration”
Indefinite Integral.
Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by 
Thus, 
Where 
is primitive of 
and C is an arbitrary constant known as the constant of integration.
Here 
is the integral sign, f(x) is the integrand, x is the variable of integration and dx is the element of integration or differential of x. The process of finding an indefinite integral of a given function is called integration of the function.
It follows from the above discussion that integration of a function f(x) means finding a function 
such that 
Basic theorems on Integration.
If f(x) and g(x) are function of x and a is a constant, then
(1) 
(2) 
(3) 
(4) 
where C is an arbitrary constant known as the constant of integration.
Fundamental Integration Formulae.
Since 
therefore based upon this definition and various standard differentiation formulae, we obtain the following integration formulae.
(1) 
(2)
(3) 
(4) 
By applying this procedure we can find out following integration
(5) 
(6) 
(7) 
(8) 
(9) 
(10) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
Methods of integration.
(1) Integration by substitution (2) Integration by parts
(3) Integration of rational algebraic functions by using partial fractions.
(1) Integration by substitution : Generally we apply this method in the following cases
(i) When integrand is a function of function : 
. Here we put 
. So that 
and in that case the integral is reduced to 
(ii) When integrand is product : of two factors such that one is the derivatives of other 
In this case we put 
=t and convert it to standard integral.
Now 
; \
(iii) Integral of a function of the form 
: We put ax + b = t and convert it into standard integral.
Now 
.
(iv) Standard form of integrals :
(a) 
put 
(b) 
, 
, put 
(c) 
, put 
(v) Standard substitution:
|
Integrand form |
Substitution |
Integrand form |
Substitution |
|
(a) |
or |
(b) |
|
|
(c) |
or |
(d) |
|
|
(e) |
|
(f) |
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|
(g) |
|
(h) |
|
or 
or 
or 
(2) Integration by parts : If u and v are two functions of 
then 
i.e., the integral of the product of two function = (first function) × (integral of second function) –integral of {(differentiation of first function) × (integral of second function)}
Note : q Integration with the help of the above rule is called the integration by parts. In the above rule there are two terms on RHS and in both the terms the integral of the second function is involved. Therefore in the product of two function if one of the two functions is not directly integral 
etc.) we take it as the first function and the remaining function is taken as the second function. If in the integral both the functions are easily integrable then the first function is chosen in such a way that the derivative of the function is a simple function and the function thus obtained under the integrable sign is easily integrable than the original function. Hence, choice of first and second function should be proper.
Note : q We can also choose the first function as the function which comes first in the word ILATE, where
I Stands for the inverse trigonometric function 
etc.
L Stands for the logarithmic function.
A Stands for the algebraic functions.
T Stands for the trigonometric functions.
E Stands for the exponential functions.
(3) Integration by partial fraction
: If 
and 
are two polynomials, then 
defines a rational algebraic function or a rational function of x.
If degree of 
degree of 
then 
is called a proper rational function.
If degree of 
degree of 
, then 
is called an improper rational function.
If 
is an improper rational function, we divide 
by 
so that the rational function 
is expressed in the form 
where 
and 
are polynomials such that the degree of 
is less than that of 
Thus, 
is expressible as the sum of a polynomial and a proper rational function.
Any proper rational function 
can be expressed as the sum of rational functions, each having a simple factor of 
Each such fraction is called a partial fraction and the process of obtaining them a called the resolution or decomposition of 
into partial fractions. The resolution of 
into partial fractions depends mainly upon the nature of the factors of 
as discussed below.
To Express 
into partial fraction
|
Factor in Denominator |
Corresponding partial fraction |
Factor in Denominator |
Corresponding partial fraction |
|
|
|
|
|
|
|
|
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Example : 
. To
find out value of A, we put 
in left hand side except 
and obtain value will be value of A in same way, 
except 
and 
except 
we get the value of B and C respectively.
Evaluations of integration when.
(1) Integrals of the form 
: To evaluate this type of integrals we express 
as the sum or difference of two squares by using the following steps.
Step I : Make the coefficient of x2 unity by taking it common.
Step II : Add and subtract the square of half of the coefficient of x.
Step III : Apply one of the formulae 
(2) Integrals of the form
: To evaluate this type of integrals, we express 
as the sum or difference of two squares by using the following steps.
Step I : Make the coefficient of 
unity by taking it common.
Step II : Add and subtract
square of half of the coefficient of x.
Step III : Apply formulae 
, 
(3) Integrals of the form 
dx : To evaluated this type of integrals we express the numerator as follows:
(Diff. of denominator) 
, where 
and 
are constants to be determined by equating the coefficients of similar terms on both sides. So we have
(4) Integrals of the form 
: To evaluate this type of integrals we express the numerator as follows: 
(Diff. of denominator) + 
Where 
and 
are constants to be determined by equating the coefficients of similar terms on both sides. So we have, 
(5)Integrals of the form : 
:
To evaluate this type of integrals we proceed as follows
Step I : Divide numerator and denominator both by
Step II : Replace
if any in denominator by 
Step III : Put
so that 
After applying these three steps the integral will reduce to the form 
dt which can be evaluated by the method discussed earlier.
(6) Integrals of the form
: To evaluate this type of integrals we proceed as follows
Step I : Put 
Step II : Replace in the numerator by sec2
.
Step III : Put 
so that 
After performing these three steps the integral reduces to the form 
which can be evaluated by the method discussed earlier.
(7) Integrals of the form
: To evaluate this type of integrals we express the numerator as follows: Numerator =
(Diff. of denominator) 
(denominator)
i.e., 
=
Where 
and 
are constants to be determined by comparing the coefficients of
and 
on both sides.
=
=
=
(8) Integrals of the form
: To evaluate integrals of the type 
we express the linear factor 
as follows: 
Where 
and 
are constants to be determined by equating the coefficients of similar terms on both sides. So we have 
The first integral on RHS is evaluated by putting 
and the second integral is evaluated by using 
.
(9) Integrals of the from 
, 
, where 
is a constant : To evaluate this type of integrals, divide the numerator and denominator by x2 and put
or 
whichever on differentiation gives the numerator of the resulting integrand.
(10) Integral of the form 
– 
: Where P and Q linear or quadratic in x, by following substitution, we can change this integral into simple form:
(i) When P and Q both are linear, put Q = t2 (ii)
When P is linear and Q is quadratic 
(iii) Both P and Q are quadratic put 
(iv)
When P is quadratic and Q is linear put 
(11) Integrals of the form
Where
m, n
are positive integers : In the integrals of the form 
the following substitutions are useful
(i) If m is odd i.e., power of 
is odd, put 
(ii) If n is odd i.e., power of 
is odd, put 
(iii) If both m and n are even, then we use De–Moivre’s theorem
Some important integrals.
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
Reduction formulae for some special cases .
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
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