Indefinate Integration

Integration Formulae
Integration By Part
Algebric Integral
Solved Examples

Integration




Integration is the reverse process of differentiation. The process of finding ƒ(x), when derivative ƒ'(x) is given is known as integration .

Integral as antiderivative

















Example
ƒ(x) = 4x2 + 6
   d/dx {ƒ(x)} = 8x
   ∫8xdx = 4x2 + c
   where c is unknown constant.

Integration Formulae


S. no. Differentiation Formulae Corresponding Integration formula
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv)
(xv)
(xvi)
Top

Integration By Part





If the integrand can be expressed as the product of two fuctions, then we use the following formula .

∫ƒ1(x)ƒ2(x)dx = ƒ1(x) ∫ƒ2(x)dx - ∫{(ƒ1'(x) ∫ƒ2(x)dx)}dx

where ƒ1(x) and ƒ2(x) are known as first and second function respectively .

In words: Integral of the product of two functions



Note : Order of ƒ1(x) and ƒ2(x) is normally decided by the rule ILATE , where

I = Inverse trigonometric
L = Logarithmic
A = Algebraic
T = Trigonometric
E = Exponential

Algebric Integral



(1)


(2)


(3)


(4)


(5)


(6)



(7)                           =


(8)                           =                              


(9)                          =
Top

Solved Examples





Example
: Write down the integral for               .

Solution

Let







[ The integral of the difference of two functions is equal to the difference of their integrals]







Example : ∫1/(8x+2)dx .

Solution
           Putting 8x+2 = t, we get
           8dx = dt
           dx = (1/8)dt

           Therefore, ∫1/(8x+2)dx = 1/8∫(1/t)dt

          = 1/8 log ‌‌‌‌‌ t ‌ + C
          = 1/8 log ‌‌‌ 8x+2 ‌ + C [Putting  t = 8x+2]


Example
:


Solution :  




                     dx = xdt
Example : Integrate the function sinx.sin2x.sin3xdx

Solution :  ∫sinx.sin2x.sin3xdx 

                  ∫(sinx.sin3x)sin2xdx

                  ∫1/2[cos2x - cos4x]sin2xdx

                  ∫1/2[cos2xsin2x - cos4xsin2x]dx

                 1/4 ∫[sin4x - (sin6x - sin2x)]dx 

              = 1/4 [∫sin4xdx - ∫sin6xdx + ∫sin2xdx]








Example : ∫xsinxdx

Solution :  Let the first function x = ƒ1(x) and
 The second function = sinx = ƒ2(x)

 ∫ƒ1(x)ƒ2(x)dx = ƒ1(x) ∫ƒ2(x)dx - ∫{(ƒ1'(x) ∫ƒ2(x)dx)}dx

 ∫xsinxdx = x ∫sinxdx - [∫(d/dx)x∫sinx dx]dx

 = x(-cosx) - ∫[1.(-cosx)]dx

 = -xcosx - ∫(-cosx)dx

 = -xcosx + sinx + C


Example
: Evaluate

Solution :

[Add and subtract 4 to convert x2 + 4x into a perfect square]













Example
: Integrate


Solution :

This is now easy to integrate:
Top
Clicky Web Analytics