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HOW TO LEARN INTEGRATION FORMULAE/FORMULAS USING TRICKS

Let us learn and remember most Important formulas of Integration , tips and tricks to learn algebraic ,most important differentiation questions for plus 2 maths, indefinite integration tricks and shortcuts trigonometric and by parts formulas in an easy and short cut manners.


Trigonometric Formulae



  1  ∫ sin x dx           =  - cos x +c  


where "c" is called constant of Integration.

The integration of sin x is  - cos x ,then divide it with the derivative of its angle. 


If we have to find the integration of  sin 2x , then we shall find it as


Step1    1st find the integration of sin x which is - cos x .

Step2    Divide it with the derivative of 2x ,which is 2, so 


∫ sin 2x dx     =  - ( cos 2x) 2 + c ,
   ∫ sin 8x dx      =  - ( cos 8x) 8 + c ,

∫ sin  3x4  dx  =   - ( cos  3x4 )   3 4  + c ,
Therefore   ∫ sin nx dx =  - ( cos nx) n +c ,






2  ∫ cos x dx          =    sin x +c  

The integration of cos x is  sin x ,then divide it with the derivative of its angle.
If we have to find the integration of  cos 2x , then we shall find it as

Step 1   1st find the integration of cos x which is sin x .

Step 2   Divide it with the derivative of 2x ,which is 2, so 


∫ cos 2x dx     =  ( sin 2x) 2 + c ,

∫ cos 8x dx     =  ( sin 8x) 8 + c ,

∫ cos  3x4  dx  = ( cos  3x4 )   3 4  + c ,

Therefore   ∫ cos nx dx =   ( sin nx) n +c ,



3  ∫ tan x dx = log |sec x| +c or - log |cos x| + c


The integration of tan x is   log |sec x| +c   or -log |cos x| +c  , then divide it with the derivative of its angle.


If we want to find the integration of tan  x2  . The integration of  tan  x2 is  log |sec x2 | +c , then divide it with the derivative of its angle.

Step 1

Find the integration of  tan  x2 ,which is log |sec  x2 | or  - log |cos  x2 |,


Step 2
 Divide it with the derivative of angle x2 ,which is 2x.

Therefore
∫ tan  x2  dx = -(1/2x) log |cos x2 | +c or (1/2x)log |sec x2 | + c

4  ∫ cot x dx = log |sin x| +c or - log |cosec x| + c


 The integration of cot x is   -log |cosec x| +c   , then divide it with the derivative of its angle.



If we want to find the integration of  cot x2  . Then integration of  cot  x2 is  -log |cosec x2 | +c  or  log |sin x2 | +c  , then divide it with the derivative of its angle.

Step 1 

 Find the integration of  cot  x2 ,which is -log |coec  x2 | or  log |sin x2 |,

Step 2

  Divide it with the derivative of angle x2 , which is 2x.   

Therefore      ∫ cot  x2  dx = -(1/2x) log |cosec x2 | +c or (1/2x ) log |sin x2 | + c



 5  ∫ sec x dx         =  log |sec x - tan x | +c 


If we want to integrate sec√x .Then 1st of all we apply the formula of integration of sec(any angle) then divide with the formula of integration of √x,So we have


 ∫  sec√x dx = ( log |sec√x - tan √x | )(2√x) +c


6 ∫ cosec x dx = - log |cosec x - cot x | +c

If we want to integrate cosec√x .Then 1st of all we apply the formula of integration of cosec (any angle) then formula of integration of √x,So we have



 ∫  cosec√x dx = - (log | cosec√x -co√x | )(2√x) +c
  

7  ∫ sec2 x dx = tan x + c


Because the derivative of tan x is sec 2 x , So the Antiderivative or Integration of sec 2 x  is tan x .




∫ sec 2 √x dx     =  (2√x ) tan √x  + c

∫ cosec 2 dx = - cot x +c



Because the derivative of  cot x is  - cosec 2 x , So the  Anti derivative or Integration of  cosec 2 x is - cot x .









 8 ∫ sec x tan x dx = sec x +c



Because the derivative of sec x is sec x tan x ,Therefore the integration of tan x sec x is sec 
x .


If we want to integrate sec√x .tan √x .Then its  integration  will be sec √x,


    sec √x tan √x   dx      =  √x   sec √x + c  


9 ∫ cosec x cot x dx = - cosec x +c


Because the derivative of cosec x is    - cosec x cot x , Therefore the integration of tan x sec x is sec x .


∫ cosec √x cot √x dx = - ( 2 √x  cosec √x ) +c 


 Also Read   WHAT IS SET, TYPES OF SETS, UNION INTERSECTION AND VENN DIAGRAMS


Algebraic Formulae


1 ∫ (constant) dx = (constant ) x +c

Integration of constant function is the constant function itself multiplied by the variable .

∫ 5 dx   = 5x  +c

2  ∫  xn  dx  = xn+1  n+1dx  + c ,

∫ x3   dx  =  x4  4  + c ,


HOW TO LEARN INTEGRATION  FORMULAE/FORMULAS USING TRICKS

   
To find the integration of function where variable "x" or f(x) has power 'n' , where "n" is any real number, we shall increase the power of "x"  by 1 and divide it with increased power.
e.g 

 ∫  x2  dx   =  {1/(2+1)} x2+1  + c

∫  (x ) 2/3   dx     =   (x ) (2/3)+1  (2/3)+1 + c ,

                  =   3(x ) 5/3  5 + c ,

∫  (ax+b ) n   dx     =   (ax+b ) n+1  a(n+1) + c ,

∫  (3x + 7 ) 2   dx     =   (3x+7 )2+1  3(2+1) + c ,

                              =   (3x+7 )3  9 + c ,

If we have to integrate sum of two functions ,then we shall integrate it separately as follows

4  ∫  [ f(x) + g(x)] dx = ∫f(x)dx +  g(x) dx + c


∫  [{ x2}3  + (2x) ]dx = ∫ { x2}3 dx +  ∫ (2x) dx

  =∫   x6dx +  ∫ 2x dx =  
 x  6+1  6+1 +(2/2)x2  + c

 =   x  7  7 x2  + c

∫  {4x2  + 3x }dx = 4x2   + ∫ 3x dx
                         =  4×(1/3)x3  (3/2)x2 +c


 ∫ [ 6x / 3x2] dx = log |  3x2 | + c

HOW TO LEARN INTEGRATION  FORMULAE/FORMULAS USING TRICKS




Memorize  these integration formulas along with differentiation in Hindi




Integration By Parts 

∫ [ f(x) g(x)] dx = f(x) ∫ g(x) dx -  {f '(x) ∫ g(x) dx}dx + c


    
∫ x sin x dx = x ∫ sin x dx - x' { ∫ sin x dx}dx + c
= x(-cos x) -  (-cos x)dx + c
 = -x cos x - sin x +c

∫ log x dx =  ∫ log x.1 dx 
= log x  - f '(log x) ( x )dx + c
= log x .1  - ∫(1/x)  x dx + c  
= log x  - ∫ 1 dx + c 
= log x  - x  + c


 Integration   Exponential Function

1  ∫  ex  dx   =  ex  + c

2 ∫  ax  dx = ax / log a    + c  

3 ∫ log x dx = x log x - x + c

4 ∫ (1/x ) dx = ln |x | + c




Exponential and Derivative Mixed Formula

   ex  [ f(x) + f '(x)] dx =ex  f(x) + c


  ex  [ sin x + cos x] dx = ex  sin x + c

Conclusion


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