**MATHS :: Lecture 10 :: ****INTEGRATION**

**INTEGRATION**

Integration is a process, which is a inverse of differentiation. As the symbol represents differentiation with respect to x, the symbol stands for integration with respect to x.

**Definition**

If then f(x) is called the integral of F(x) denoted by . This can be read it as integral of F(x) with respect to x is f(x) + c where c is an arbitrary constant. The integral is known as **Indefinite integral and the function F(x) as integrand.**

Integration by parts Examples I

Integration by parts Examples II

Integration by parts Examples III

** Formula on integration **

**1). +c **( n ¹-1)** **

**2). +c **

**3). = x+c **

**4). +c **

**5). **dx = ex **+c **

**6). **

**7). **

** 8). = **c x + d** **

** 9). +c **

**10). +c **

**11). +c **

**12). +c **

**13). **

**14). **

**13). +c **

**14). +c **

**15). +c **

**16). **

**Definite integral**

If f(x) is indefinite integral of F(x) with respect to x then the Integral is called definite integral of F(x) with respect to x from x = a to x = b. Here a is called the Lower limit and b is called the Upper limit of the integral.

= = f(Upper limit ) - f(Lower limit)

= f(b) - f(a)

**Note **

While evaluating a definite integral no constant of integration is to be added. That is a definite integral has a definite value.

###### Method of substitution

**Method –1**

**Formulae for the functions involving (ax + b)**

Consider the integral

I = -------------(1)

Where a and b are constants

Put a x + b = y

Differentiating with respect to x

a dx + 0 = dy

Substituting in (1)

I = +c

=+c

=+c

=+ c

Similarly this method can be applied for other formulae also.

**Method II**

**Integrals of the functions of the form**

put =y,

Substituting we get

I = and this can be integrated.

**Method –III**

**Integrals of function of the type**

when n ¹ -1, put f(x) = y then

\ =

=

=

when n= -1, the integral reduces to

putting y = f(x) then dy = f1(x) dx

\=log f(x)

**Method IV**

# Method of Partial Fractions

Integrals of the form

**Case.1**

If denominator can be factorized into linear factors then we write the integrand as

the sum or difference of two linear factors of the form

###### Case-2

In the given integral the denominator ax2 + bx + c can not be factorized into linear factors, then express ax2 + bx + c as the sum or difference of two perfect squares and then apply the formulae

**Integrals of the form**

**Write denominator as the sum or difference of two perfect squares**

=** or or **

**and then apply the formula**

** **= log(x+

** **= log(x+

** = **

**Integration by parts**

If the given integral is of the form then this can not be solved by any of techniques studied so far. To solve this integral we first take the product rule on differentiation

=u +v

Integrating both sides we get

dx= ( u +v )dx

then we have u v=+

re arranging the terms we get

= uv- This formula is known **as integration by parts formula**

Select the functions u and dv appropriately in such a way that integral can be more easily integrable than the given integral

Application of integration

The area bounded by the function y=f(x), x=axis and the ordinates at x=a x=b is given by

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