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 +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

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

Step 2.    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 |cosec   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 -cot √x | )(2√x) + c

  

7  ∫ sec² x dx = tan x + c

Because the derivative of tan x is sec²  x , So the Anti derivative or Integration of sec² x  is tan x .

∫ sec² √x dx  =  (2√x ) tan √x  + c

∫ cosec²  dx = - cot x + c

Because the derivative of  cot x is  - cosec² 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      =  2 √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  

 

Integration of Trigonometric Functions

S N	f(x)	∫dx
1	sin x	-cos x + c
2	cos x	sin x  + c
3	tan x	ln|sec x| + c
4	cot x	ln|sin x| + c
5	sec x	ln|sec x + tan x |+ c
6	cosec x	-ln|cosec x - cot x |+ c

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 ,

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 ) ^⅔   dx     =   (x ) ^{(⅔)+1  (⅔)+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

 =   x⁷    ⁷ + x²  + c

∫  {4x²  + 3x }dx = ∫4x²   + ∫ 3x dx

                         =  4×(1/3)x³ +  (3/2)x² + c

 ∫ [ 6x / 3x²] dx = log |3x² | + c

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 ∫x  - ∫f '(log x) ( x )dx + c

= log x .1  - ∫(1/x)  x dx + c  

= log x  - ∫ 1 dx + c 

= log x  - x  + c

Also Read How To Understand Binary operations , Relations And Functions , Commutative  and  Associative  

 Integration   Exponential Function

1  ∫  e^x  dx   =  e^x  + c

2 ∫  a^x  dx = a^x / log a    + c  

3 ∫ log x dx = x log x - x + c

4 ∫ (1/x ) dx = ln |x | + c

Exponential and Derivative Mixed Formula

  ∫ e^x  [ f(x) + f '(x)] dx =e^x  f(x) + c

 ∫ e^x  [ sin x + cos x] dx = e^x  sin x + c

Conclusion

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