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

Thanks for devoting your valuable time for this post "How To learn Integration  Formulae / Formulas very   easily" of my blog . If you liked this  blog/post, Do Follow me on my blog and share this post with your friends . We shall meet again with new post  ,till then Good Bye.

Read More

HOW TO SOLVE HARD AND IMPOSSIBLE PUZZLES PART 2

 we are going to discuss some of the puzzles, challenges or brain teaser which are viral in social media such as Face book ,whats app and twitter    etc. which are based on Mathematics directly or indirectly . You have seen such types of puzzles mentioned 99.9% Fail , Solve if you are genius like this........

Let us try to discuss and solve them. 

PUZZLE # 1

Which number should replace the ?

  4 ,  9 ,  19  ,  39  , 79   ?

when we subtract   4 from  9  we get   5 i.e     9 -  4  =  5 

when we subtract   9 from 19 we get 10 i.e   19 -  9  = 10 

when we subtract 19 from 39 we get 20 i.e.  39 - 19 = 20

when we subtract 39 from 79 we get 40 i.e.  79 - 39 = 40

Now analyse the trends of answers ,we are getting double of the previous number as a result every time .Similarly if we follow the same trend from 79 to next term ,then we shall get 80 as answer. So to get 80 as answer "Add 80 to 79 to get 159 ". when we subtract 79 from 159 ,we get the double of the previous result .

So final answer is  159 - 79 = 80.

  4 ,  9 ,  19  ,  39  , 79  , 159

 Therefore  " ? " will replace 159 as Answer.

PUZZLE # 2

But before we proceed to discuss the Puzzle-1, let us discuss some of the necessary keywords/definitions which have been used in  it.

Even Number

When any Number is divisible by  2 then that number is called An Even number. If we take 2, 16, 84, 22, 20, 100 etc then they are Even Numbers as they can be divided by 2.

Let us learn Multiplication , division , arithmetic and simplification short cut ,tips and  tricks after buying this Book of Magical Mathematics.

Odd Number

When any number is NOT divisible by  2 then that number is called An ODD number .If we take 21, 15, 49, 21, 57, 101 etc then they are Odd  Numbers as they can NOT  be divided by 2.

Multiple of 3 

An number which is divisible by 3 or which comes in the  table of 3 is called  multiple of 3. if we take 57, 102, 69, 9, 6, 21, 39 etc numbers , as these numbers comes in the table of 3 or these are divisible by 3 , so they are Multiple of 3.

Prime Number

Any number which has only two Factors 1 and the numbers itself is called Prime number. 1 is not the the prime number, 2 ( It is the 1st prime number ) , 3, 5, 7, 11, 13, 17, 19, 23  etc  are examples of Prime numbers as these numbers have only two factors 1 and the number itself.

Square Number

Any number whose square root is a Natural  number is called square Number . such as  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 etc.

 Factors of 360

All those numbers which can divide 360 are called factors of 360 .Here We can take such numbers as 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 18, 36 etc

How to Solve 

Have you still not visited "HOW TO SOLVE HARD AND IMPOSSIBLE PUZZLES  PART 1"

PUZZLE # 3

When we multiply   2   with  4,  we get     8
When we multiply   3   with  6,  we get    18
When we multiply   4   with  8,  we get    32
When we multiply   5   with 10, we get    50
When we multiply   6   with 12, we get    72
When we multiply   7   with 14, we get    98
When we multiply   8   with 16, we get    128
When we multiply   9   with 18, we get    162
When we multiply  10  with 20, we get    200

Analyse the Answers above  , we get  4, 6, 8, 10, 12, 14, 16, 18 and 20. Is there any relation between them. Of course, Because every answer is 2 more than the previous Answer. So Last number  should be 200. so in order to get answer 200 we have to multiply  20 with 10.

So       If 2  =  8  

               3  = 18 

               4  = 32

               5  = 50 
               6  = 72  
               7  = 84
               8  = 128
               9  = 172 
              10 = 200 

Then    10 = 200

In order to get any number any of any term  based upon this trends we can use the formula  2n² , where "n "  is the number appearing on left side of the Puzzle.

So Final Answer will be 200

Here is the Solution of the Puzzle asked in "HOW TO SOLVE HARD AND IMPOSSIBLE PUZZLES PART 3" as follows.

There are two numbers 'a' and 'b' in 1st and 2nd columns then 3rd column has been  occupied by their combination as follows

a*b = (a²  + b² ) - 1,

1*3 = (1²  + 3² ) - 1 = ( 1 +   9) -1  = 10 - 1 =  9

3*4 = (3²  + 4² ) - 1 = ( 9 + 16) -1  = 25 - 1 = 24

2*5 = (2²  + 5² ) - 1 = ( 4 + 25) -1  = 29 - 1 = 28

4*5 = (4²  + 5² ) - 1 = (16 + 25) -1 = 41 - 1 = 40

So 40 will replace  ?  in this puzzle   

Conclusion

Thanks for devoting your valuable time for this post  "How   to solve various Hard and impossible puzzles ,Quizzes ,  Brain Teasers and challenges " of my blog . If you liked this this blog/post,  Do Follow me on my blog and share this post with your friends . We shall meet again   in next post with solutions of most interesting and mind blowing puzzles ,till then Good Bye.

                                                                       

Read More