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BASIC SUBSTITUTION METHOD

COMPLETING THE SQUARE

TRIGONOMETRIC IDENTITIES

IMPROPER FRACTION

SEPARATNG FRACTION

MULTIPLYING BY A FORM OF 1

ELIMINATING SQUARE ROOTS

INTEGRATION BY PARTS

TRIGONOMETRIC INTEGRALS

INTEGRATION OF RATIONAL F(X) BY PARTIAL FRACTION

TRIGONOMETRIC SUBSTITUTIONS

IMPROPER INTEGRALS

RUNNING CHAIN RULE

TRIGONOMETRIC F(X)

EXPONENTIAL F(X)

GENERAL EXPONENTIAL F(X)

INVERSE TRIGONOMETRIC FUNCTIONS

HYPERBOLIC F(X)

INVERSE HYPERBOLIC F(X)

NATURAL LOGARITHM

DEFINITE

INDEFINITE

F - D - (I+ LOWER & UPPER) <ul><li>FIND "U" , THEN DIFFERENTIATE

"U" , INTEGRATE BUT DON'T

FORGET TO PUT LOWER AND

UPPER LIMIT</li></ul>

F - D - (I THEN +C) <ul><li>FIND "U", THEN DIFFERENTIATE "U", THEN INTEGRATE AND +C </li></ul>

INDEFINITE

DEFINITE

M - S - <ul><li>MEMORIZE AL FORMULAS AND SIMPLIFY THE EQUATION</li></ul>

I IF CAN <ul><li>INTEGRATE IF WE CAN INTEGRATE IT</li></ul>

IF CANNOT I . THEN F - D - I <ul><li>IF CANNOT INTERGRATE , THEN , FIND U, DIIFF U AND

INTEGRATE U </li></ul>

M - S <ul><li>MEMORIZE ALL FORMULAS AND SIMPLIFY THE EQUATION</li></ul>

I IF CAN

IF CANNOT I THEN F-D-I <ul><li>IF CANNOT INTEGRATE, THE FIND U

DIFFERENTIATE U

INTEGRATE U</li></ul>

IF LINEAR FX THEN I <ul><li>I FOR INTEGRATE</li></ul>

IF NOT LINEAR FX

S-F-D-I <ul><li>SIMPLIFY IT THEN IDENTIFY "U" -&gt; DIFFERENTIATE "U" --&gt; INTEGRATE IT IF NO UPPER OR LOWER LIMITS THEN PUT +C</li></ul>

IF LINEAR FX THEN I

IF NOT LINEAR FX

S-F-D-I <ul><li>SIMPLIFY IT --&gt; FIND "U" --&gt; DIFFERENTIATE "U" --&gt; INTEGRATE "U" , IF NO UPPER OR LOWER LIMIT THEN +C</li></ul>

M - S - F - D - I <ul><li>MEMORIZE ALL FORMULAS --&gt; SIMPLIFY IT --&gt; FIND "U" --&gt; DIFFERENTIATE "U" --&gt; INTEGRATE </li></ul>

M - S - F - D- I <ul><li>MEMORIZE ALL FORMULAS --&gt; SIMPLIFY IT --&gt; FIND "U" --&gt; DIFFERENTIATE "U" --&gt; INTEGRATE</li></ul>

S - F - D - I <ul><li>SIMPLIFY IT --&gt; FIND "U" --&gt; DIFFERENTIATE "U" --&gt; INTEGRATE</li></ul>

NO SUBSTITUTION AVAILABLE

F-I <ul><li>FACTORIZE THE DENOMINATOR -&gt; INTEGRATE THE F(X)</li></ul>

SUMMATION OF 2 TERM POWER 2 & DIFFERENT ANGLE

M-E-S-I <ul><li>MEMORIZE ALL FORMULAS -&gt;

EXPAND IT -&gt; SIMPLIFY BY USING ANY FORMULA -&gt; INTEGRATE F(X)</li></ul>

DEGREE OF NUMERATOR > DENOMINATOR

L D M - S -I <ul><li>USE LONG DIVISION METHOD

-&gt; SIMPLIFY IT -&gt; INTEGRATE F(X)</li></ul>

CAN BE SEPARATED

SE - S- I <ul><li>SEPARATE THE F(X) -&gt; SIMPLIFY -&gt; INTEGRATE</li></ul>

SINGLE F(X) CANNOT INTEGRATE

C - S - I <ul><li>CONJUGATE THE TERM -&gt; SIMPLIFY -&gt; INTEGRATE F(X)</li></ul>

TRIGO F(X) IN SQUARE ROOT

S - A - I <ul><li>SIMPLIFY -&gt; KNOW THE AREA

-&gt; INTEGRATE</li></ul>

F(X) IS NOT DIFFERENTIAL COEFFICIENT

TABULAR INTEGRATION

SPECIAL CASE

UV( PRODUCT RULE)

u can differentiated repeatedly until 0

I L ATE

u cannot differentiated repeatedly until 0 but we differentiate it until get (-) from the f(x)

I L A T E <ul><li>I - INVERSE TRIGO

L -LOGARITHMIC

A- ALGEBRAIC

T- TRIGONOMETRIC

E- EXPONENTIAL/GENERAL EXPONENTIAL</li></ul>

WHEN WE HAVE DIFFERENT TRIGONOMETRIC ANGLES AND DIFFERENT POWER

PRODUCT OF POWERS OF SINES AND COSINES

PRODUCT OF SINES AND COSINES

T SUBSTITUTION

LP - SE INTO EVEN POWER - E - S - I <ul><li>FIND THE LOWER POWER --&gt; SEPARATE IT INTO EVEN POWER --&gt; EXPAND --&gt; SUBSTITUTE USING HALF ANGLE --&gt; INTEGRATE</li></ul>

M - S - I <ul><li>MEMORIZE ALL FORMULAS --&gt; SUBSTITUTE THE FORMULA --&gt; INTERGATE</li></ul>

WHEN COS AND SIN LOCATED AT DENOMINATOR

Use T substitution by drawing triangle --> identify sin and cos --> Substitute using any suitable formula --> integrate

NONREPEATED

REPEATED

Separate the f(x)---> label the separated f(x) -> integrate it

Heavysight cover up method

Separete the f(x) --> label according by repeated f(x) -> Integrate

When we replace the variable of integration with trigonometric f(x)

F - D - U -I - RA <ul><li>FIND THE SUBSTITUTION FOR x --&gt; DIFFERENTIATE dx --&gt; USE CORRECT VALUE OF ANGLE FOR SUBSTITUTION --&gt;INTEGRATE THE TRIGONOMETRIC INTEGRAL --&gt; DRAW AN APPRORIATE REFERENCE TRIANGLE ANGLE</li></ul>

Find the substitute for x --> differentiate dx --> specify the selected values of angle -->integrate the trigonometric integral --> draw an appropriate reference triangle

WHEN SUBSTITUTE LOWER & UPPER LIMT THE ANSWER WILL BE UNDEFINED

UPPER AND LOWER LIMIT INFINITY

Separate the equation --> exist a new limit--> integrate ---> substitute the lower and upper limit

Exist a new limit---> integrate---> Identify the suitable graph---> substitute the lower & upper limit

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