Indefinite Integral

Indefinite integral

In many practical problems, the derivative of a function is given and we are required to find the function which has the given derivative. The process of finding this function is called integration.


Definition: Basic Concept
Let F(x) be a differentiable function of x such that . Then F(x) is called the integral of f(x). Symbolically, it is written as f(x) dx = F(x). f(x), the function to be integrated, is called the integrand. F(x) is also called the anti-derivative (or primitive function) of f(x).


Constant of Integration:

As the differential coefficient of a constant is zero, we have

.Therefore, f(x)dx = F(x) + c.

The constant c is called the constant of integration and can take any real value. By assigning different value to c, we obtain different values of the integral.



Basic formulae:

Antiderivatives or integrals of some of the widely used functions (integrands) are given below:

, n - 1





(a > 0)













or -
















Standard Formulae:

















Properties of indefinite integration:

Let f (x) and g (x) be two integrable functions. Then the following results hold:

i) where a is an arbitrary constant.

ii) .

iii) .

iv) If f(u)du = F(u) + c, then f(ax + b) dx =



Methods of Integration

If the integrand is not a derivative of a known function, then the corresponding integrals cannot be found directly. In order to find the integrals of complex problems, generally four methods are used so that the integral is reduced to the standard form.

Integration by substitution or by change of the independent variable.

Integration by parts.

Integration by using trigonometric identities.

Integration by partial fractions.

Integration By Substitution



Direct Substitutions:

If the integral is of the form f(g(x)) g'(x) dx, then put g(x) = t, provided exists.

g (x) = t so that the integral become.


The problem of integrating with respect to x is transformed to integrating f (t) with respect to t and = k (t) + c = k (g (x)) + c.



Standard Substitutions:

For terms of the form x² + a² or , put x = a tan or a cot.

For terms of the form x² - a² or , put x = a sec or a cosec.

For terms of the form a² - x² or , put x = a sin or a cos.

If both , are present, then put x = a cos.

For the type, put x = a cos²q + b sin²q.

For the type or , put the expression with in the brackets = t.

For the type (n N, n > 1),

put .

For , n₁, n₂ N (and > 1), again put (x + a) = t(x + b).


Indirect Substitutions: If the integrand is of the form f(x).g(x), where g(x) is a function of the integral of f(x), then put integral of f(x) = t.




Derived Substitutions:

Some times it is useful to write the integral as a sum of two related integrals, which can be evaluated by making suitable substitutions. Examples of such integrals are:


A. Algebraic Twins









B. Trigonometric Twins
, ,

, .



Integrals of trigonometric functions:

(i). .

(ii)

(iii) = log |secx + tanx| + c.

(iv) = log |cosec x - cot x| + c.



Integrals of some particular functions:

Remarks:

If f (x) is a differentiable function of x, then

(i)

(ii)

(iii) by writing f (x) = t.

Integration by Parts If u and v be two functions of x, then integral of the product of these two functions is given by: .

This method is mainly used when the integrand is the product of two functions.


Note:In applying the above rule care has to be taken in the selection of the first function(u) and the second function (v). Normally we use the following methods:

(i) If in the product of the two functions, one of the functions is not directly integrable then we take it as the first function and the remaining function is taken as the second function.

(ii) If there is no other function, then unity is taken as the second function

(iii) If both of the function are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually we use the following preference order for the first function. (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent). In this order, the function on the left is always chosen as the first function. This rule is called as ILATE

(iv) Sometimes, it may be necessary to integrate by parts more than once to complete the integration process


.
An important result:

In the integral if g(x) can be expressed as

g(x) = f(x) + f'(x) then= e^x f(x) + c.



Integration by Partial Fractions

A function of the form P(x)/Q(x), where P(x) and Q(x) are polynomials, is called a rational function. Consider the rational function .

The two fractions on the RHS are called partial fractions. To integrate the rational function on the LHS, it is enough to integrate the two fractions on the RHS, which are easily

integrable. This is known as method of partial fractions.



Note:
Before proceeding to write a rational function as a sum of partial fractions, we should ascertain that it is either a proper rational fraction or is rewritten as one.



Algebraic Integrals

I = .

Here ax² + bx + c = a= .

Substituting, we have

I = .


I = .


Using the substitution as above, we have


I =


I = dx is reduced to


I = .





Integration of Irrational Algebraic Fractions:

(i). Irrational functions of (ax+b)^1/n and x can be easily evaluated by the substitution

tⁿ = ax+ b. Thus.


(ii). For , we substitute, x -k = 1/t.

This substitution will reduce the given integral to -.


(iii). , so that

.

Now the substitution C + Dt² = u² reduces it to the form.


(iv). To evaluate

we write, ax² +bx +c = A₁ (dx +e) ( 2fx +g) +B₁( dx +e) +C₁

where A₁, B₁ and C₁ are constants which can be obtained by comparing the coefficients, of like terms on both sides. And given integral will reduce to the form

A₁.



Trigonometric Integrals

Integrals of the form R (sinx, cosx) dx:

Here R is a rational function of sin x and cos x. This can be translated into integrals of a rational function by the substitution: tan(x/2) = t, This is the so called universal substitution. In

this case

.

(a) If R(-sin x, cos x) = -R(sin x, cos x), substitute cos x = t.

(b) If R(sin x, -cos x) = -R(sin x, cos x), substitute sin x = t.

(C) If R(-sin x, - cos x) = R(sin x, cos x), substitute tan x = t.



Integrals of the form:




Rule for (i) :

In this integral express numerator as l (denominator) + m(d.c. of denominator) + n. Find l, m, n by comparing the coefficients of sinx, cosx and constant term and split the integral into

sum of three integrals

.


Rule for (ii) :

Express numerator as l (denominator) + m(d.c. of denominator) and find l and m as above.

For I =, we write a = r cos , b = r sin

and get I =

= |sec (x - ) + tan (x - )| + c

where r = and = tan^-1.



Integration of the type:
(sin^Mxcos^Nx)dx

M & N natural numbers.

If one of them is odd, then substitute for term of even power.

If both are odd, substitute either of the term.

If both are even, use trigonometric identities only.