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Title: Calculus
Description: Indefinite Integration along with practice problems.
Description: Indefinite Integration along with practice problems.
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Integration
Introduction
Differential of y = f(x)
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Antiderivative of a periodic function need not
be periodic function
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f(x) = cos x + 1 is periodic but
= sin x + x + C is aperiodic
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(a)
(b)
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= tan–1x+C
= sin–1x+C
= sec–1x+C
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find f(x) if
f ' (sin2x) = cos2x for all x, f(1) = 1
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Compute g(4)
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By substitution
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Partial fraction
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(kuturputur)
Substitution
Or
Or
Or
Start with y = f(x)
Illustrations
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(3)
Both even use T-identities to manipulate
(4)
If m & n are rational numbers &
is
negative integer or m & n is negative even
integer then, substitute
Also create
derivatives
Examples
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Integration By Parts
Rules :–
(i)
Choose 2nd Function which is easily integrable
(ii)
Choose 1st & 2nd functions such that after by
parts Complexity of 2nd term reduces as
compared to original integration
(iii) Note sometimes 1 is taken as a function
Examples
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If the primitive of the function
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Partial Fraction
Case I :
If
P1(x), P2(x) are polynomials
If degree of P1 > P2 Divide & Move to case (2)
Case II :
Degree of P1(x) < Degree of P2(x)
(a) P2 is linear in x
For Example :
Let
, B = 4,
(b)
Denominator is linear factor of x
(i)
(b)
Denominator is quadratic in x (Not Factorized)
(ii)
Important Concepts
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Miscellaneous
Kuturputur
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Integration of Irrational
Algebraic Function
Working Rule :
Put L2 = t
2
Working Rule :
Put L1 = t2
Working Rule :
Put L = 1/t
Case 1 :
For
Q1
D>0
Case 2 :
For
Q1
D=0
Case 3 :
For
Q1
D<0
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Title: Calculus
Description: Indefinite Integration along with practice problems.
Description: Indefinite Integration along with practice problems.