TECHNIQUES OF INTEGRATION

TECHNIQUES OF INTEGRATION

1. BASIC SUBSTITUTION METHOD

   • RUNNING CHAIN RULE
     • DEFINITE
       • F - D - (I+ LOWER & UPPER)
         • FIND "U" , THEN DIFFERENTIATE "U" , INTEGRATE BUT DON'T FORGET TO PUT LOWER AND UPPER LIMIT
     • INDEFINITE
       • F - D - (I THEN +C)
         • FIND "U", THEN DIFFERENTIATE "U", THEN INTEGRATE AND +C 
   • TRIGONOMETRIC F(X)
     • INDEFINITE
       • M - S -
         • MEMORIZE AL FORMULAS AND SIMPLIFY THE EQUATION
         • I IF CAN
           • INTEGRATE IF WE CAN INTEGRATE IT
         • IF CANNOT I . THEN F - D - I
           • IF CANNOT INTERGRATE , THEN , FIND U, DIIFF U AND  INTEGRATE U 
     • DEFINITE
       • M - S
         • MEMORIZE ALL FORMULAS AND SIMPLIFY THE EQUATION
         • I IF CAN
         • IF CANNOT I THEN F-D-I
           • IF CANNOT INTEGRATE, THE FIND U DIFFERENTIATE U INTEGRATE U
   • EXPONENTIAL F(X)
     • IF LINEAR FX THEN I
       • I FOR INTEGRATE
     • IF NOT LINEAR FX
       • S-F-D-I
         • SIMPLIFY IT THEN IDENTIFY "U" -> DIFFERENTIATE "U" --> INTEGRATE IT IF NO UPPER OR LOWER LIMITS THEN PUT +C
   • GENERAL EXPONENTIAL F(X)
     • IF LINEAR FX THEN I
     • IF NOT LINEAR FX
       • S-F-D-I
         • SIMPLIFY IT --> FIND "U" --> DIFFERENTIATE "U" --> INTEGRATE "U" , IF NO UPPER OR LOWER LIMIT THEN +C
   • INVERSE TRIGONOMETRIC FUNCTIONS
     • M - S - F - D - I
       • MEMORIZE ALL FORMULAS --> SIMPLIFY IT --> FIND "U" --> DIFFERENTIATE "U" --> INTEGRATE 
   • HYPERBOLIC F(X)
     • M - S - F - D - I
       • MEMORIZE ALL FORMULAS --> SIMPLIFY IT --> FIND "U" --> DIFFERENTIATE "U" --> INTEGRATE 
   • INVERSE HYPERBOLIC F(X)
     • M - S - F - D- I
       • MEMORIZE ALL FORMULAS --> SIMPLIFY IT --> FIND "U" --> DIFFERENTIATE "U" --> INTEGRATE
   • NATURAL LOGARITHM
     • S - F - D - I
       • SIMPLIFY IT --> FIND "U" --> DIFFERENTIATE "U" --> INTEGRATE

2. COMPLETING THE SQUARE

   • NO SUBSTITUTION AVAILABLE
     • F-I
       • FACTORIZE THE DENOMINATOR -> INTEGRATE THE F(X)

3. TRIGONOMETRIC IDENTITIES

   • SUMMATION OF 2 TERM POWER 2 & DIFFERENT ANGLE
     • M-E-S-I
       • MEMORIZE ALL FORMULAS -> EXPAND IT -> SIMPLIFY BY USING ANY FORMULA -> INTEGRATE F(X)

4. IMPROPER FRACTION

   • DEGREE OF NUMERATOR > DENOMINATOR
     • L D M - S -I
       • USE LONG DIVISION METHOD  -> SIMPLIFY IT -> INTEGRATE F(X)

5. SEPARATNG FRACTION

   • CAN BE SEPARATED
     • SE - S- I
       • SEPARATE THE F(X) -> SIMPLIFY -> INTEGRATE

6. MULTIPLYING BY A FORM OF 1

   • SINGLE F(X) CANNOT INTEGRATE
     • C - S - I
       • CONJUGATE THE TERM -> SIMPLIFY -> INTEGRATE F(X)

7. ELIMINATING SQUARE ROOTS

   • TRIGO F(X) IN SQUARE ROOT
     • S - A - I
       • SIMPLIFY -> KNOW THE AREA  -> INTEGRATE

8. INTEGRATION BY PARTS

   • F(X) IS NOT DIFFERENTIAL COEFFICIENT
     • TABULAR INTEGRATION
       • u can differentiated repeatedly until 0
         • I L ATE
     • SPECIAL CASE
       • u cannot differentiated repeatedly until 0 but we differentiate it until get (-) from the f(x)
     • UV( PRODUCT RULE)
       • I L A T E
         • I - INVERSE TRIGO L -LOGARITHMIC A- ALGEBRAIC T- TRIGONOMETRIC E- EXPONENTIAL/GENERAL EXPONENTIAL

9. TRIGONOMETRIC INTEGRALS

   • WHEN WE HAVE DIFFERENT TRIGONOMETRIC ANGLES AND DIFFERENT POWER
     • PRODUCT OF POWERS OF SINES AND COSINES
       • LP - SE INTO EVEN POWER - E - S - I
         • FIND THE LOWER POWER --> SEPARATE IT INTO EVEN POWER --> EXPAND --> SUBSTITUTE USING HALF ANGLE --> INTEGRATE
     • PRODUCT OF SINES AND COSINES
       • M - S - I
         • MEMORIZE ALL FORMULAS --> SUBSTITUTE THE FORMULA --> INTERGATE
     • T SUBSTITUTION
       • WHEN COS AND SIN LOCATED AT DENOMINATOR
         • Use T substitution by drawing triangle --> identify sin and cos --> Substitute using any suitable formula --> integrate

10. INTEGRATION OF RATIONAL F(X) BY PARTIAL FRACTION

    • NONREPEATED
      • Separate the f(x)---> label the separated f(x) -> integrate it
      • Heavysight cover up method
    • REPEATED
      • Separete the f(x) --> label according by repeated f(x) -> Integrate

11. TRIGONOMETRIC SUBSTITUTIONS

    • When we replace the variable of integration with trigonometric f(x)
      • F - D - U -I - RA
        • FIND THE SUBSTITUTION FOR x --> DIFFERENTIATE dx --> USE CORRECT VALUE OF ANGLE FOR SUBSTITUTION -->INTEGRATE THE TRIGONOMETRIC INTEGRAL --> DRAW AN APPRORIATE REFERENCE TRIANGLE ANGLE
      • Find the substitute for x --> differentiate dx --> specify the selected values of angle -->integrate the trigonometric integral --> draw an appropriate reference triangle

12. IMPROPER INTEGRALS

    • WHEN SUBSTITUTE LOWER & UPPER LIMT THE ANSWER WILL BE UNDEFINED
      • Separate the equation --> exist a new limit--> integrate ---> substitute the lower and upper limit
    • UPPER AND LOWER LIMIT INFINITY
      • Exist a new limit---> integrate---> Identify the suitable graph---> substitute the lower & upper limit

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