Techniques of Integration

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hamidymuhammad
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Techniques of Integration
1 basic substitution method
1.1 1) identify the U
1.2 2) differentiate U and change the question to the form of U
1.3 3) integrate as usual
2 completing the square,CSM (when quadratic function at denominator)
2.1 1) simplify denominator using CSM
2.2 2) integrate the equation
2.3 *answer usually in the form of inverse trigo.
3 trigonometric identities (when there is summation of 2 terms ^2 @ different angle)
3.1 1) expand the equation
3.2 2) apply the trigonometric identitiesl
3.3 3) make sure every term can be integrated
3.4 4) integrate each term as usual.
4 improper fraction (when degree of numerator>/= degree of denominator)
4.1 1) use long division method to simplify the equation
4.2 2) integrate the simplified eq. as usual
5 separating fractions (when the fraction can be separated)
5.1 1) separate the function into 2 eq. with same denominator.
5.2 2) simplify then integrate each term
6 multiplying by a form of 1
6.1 1) multiply by a form of 1 or its conjugate
6.2 2) simplify the expression
6.3 3) integrate the expression as usual.
7 eliminating square roots(when trigo function is in the square root)
7.1 1) simplify to a squared trigonometric form
7.2 2) eliminate the square root
7.3 3) integrate as usual
8 integration by parts
8.1 in the form of
8.1.1 1) identify u and dv
8.1.1.1 use ILATE
8.1.2 2) substitute into the form
8.2 tabular integration
8.2.1 1) diff u until becomes 0
8.2.2 2) combine the product of fn
9 integration by partial fractions
9.1 non-repeated linear factors
9.1.1 1) simplify the deniminator
9.1.2 2) separate the denominator
9.1.3 3) find A and B
9.1.3.1 use Heaviside "cover up" method
9.2 repeated linear factors
9.3 irreducible quadratic factors
9.4 improper fraction
10 trigonometric integrals
10.1 when the trigo fn is to the power of an even no.
10.1.1 use half angle formula then integrate using basic subs. method
10.2 when trigo fn is to the power of an odd no.
10.2.1 1) release one of the factor. convert the remaining using identities.
10.2.2 2) integrate each term.
10.3 reduction formula
10.3.1 to integrate sin^n x and cos^n x
10.4 product of powers of sines and cosines.
10.4.1 both odd
10.4.2 one odd one even
10.4.3 both even
10.5 t-substitution (t=tan x)
10.5.1 1) change the eq. in terms of t
10.5.2 2) integrate the fn
10.5.3 3)convert back the t into tan x
11 trigonometric substitution
11.1 1)convert the variable in terms of theta
11.2 2) integrate as usual
11.3 3) change the theta back into variable form
12 improper integral
12.1 1) sketch the graph
12.2 2) change into limit
12.3 3) integrate the equation
12.3.1 By: Hamidy and Anas (set7)
12.4 4) solve the limit.