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Title: class 12 integration
Description: A basic(but extensive) review of all integrtation concepts for class 12 mathematics ,including formulaes
Description: A basic(but extensive) review of all integrtation concepts for class 12 mathematics ,including formulaes
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Mathematics Notes for Class 12 chapter 7
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Then, the collection of all its primitives is called the indefinite
integral of f(x) and is denoted by ∫f(x)dx
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If
d/dx {φ(x)) = f(x), ∫f(x)dx = φ(x) + C, where C is called the constant of integration or arbitrary
constant
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Geometrical Interpretation of Indefinite Integral
If d/dx {φ(x)} = f (x), then ∫f(x)dx = φ(x) + C
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The graphs of these functions give us an infinite family
of curves such that at the points on these curves with the same x-coordinate, the tangents are
parallel as they have the same slope φ'(x) = f(x)
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e
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Comparison between Differentiation and Integration
(i) Both differentiation and integration are linear operator on functions as
d/dx {af(x) ± bg(x)} = a d/dx{f(x) ± d/dx{g(x)}
and ∫[a
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g(x)dx = a ∫f(x)dx ± b ∫g(x)dx
(ii) All functions are not differentiable, similarly there are some function which are not
integrable
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(iv) Geometrically derivative of a function represents slope of the tangent to the graph of
function at the point
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Rules of Integration
Method of Substitution
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Basic Formulae Using Method of Substitution
If degree of the numerator of the integrand is equal to or greater than that of denominator
divide the numerator by the denominator until the degree of the remainder is less than that of
denominator i
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,
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(Numerator / Denominator) = Quotient + (Remainder / Denominator)
Trigonometric Identities Used for Conversion of Integrals into the Integrable Forms
Standard Substitution
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Special Integrals
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Important Forms to be converted into Special Integrals
(i) Form I
(ii) Form II
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Put px + q = λd / dx (ax2 + bx + c) + mu;
Now, find values of λ and mu; and integrate
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(iv) Form IV
After dividing both numerator and denominator by x2, put x – a2 / x = t or x + (a2 / x) = t
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(b) Rreplace sec2x, if any in denominator by 1 + tan2 x
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ncerthelp
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Put x + 1/x = t or x – 1/x = t as the case may be
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ncerthelp
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If f(x) and g(x) be two integrable
functions, then
(i) We use the following preferential order for taking the first function
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In short we write it HATE
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(iii) If only one function is there, i
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, ∫log x dx, then 1 (unity) is taken as second function
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Integral of the Form
Integration Using Partial Fractions
(i) If f(x) and g(x) are two polynomials, then f(x) / g(x) defines a rational algebraic function of
x
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(ii) If degree of f(x) ≥ degree of g(x), then f(x) /g(x) is called an improper g(x) rational
function
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(iv) Any proper rational function f(x) / g(x) can be expressed as the sum of rational functions
each having a simple factor of g(x)
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13 | P a g e
Shortcut for Finding Values of A, B and C etc
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When g(x) is expressible as the product of non-repeated line factors
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Case II
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14 | P a g e
Here, all the constant cannot be calculated by using the method in Case I
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e
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Integration of Irrational Algebraic Function
Irrational function of the form of (ax + b)1/n and x can be evaluated by substitution (ax + b) = tn,
thus
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15 | P a g e
Integrals of the Type (bxm + bxn)P
Case I
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Case II
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Case III
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16 | P a g e
Integration of Hyperbolic Functions
∫sinh x dx = cosh x + C
∫cosh x dx = sinh x + C
∫sech2x dx = tanh x + C
∫cosech2x dx = – coth x + C
∫sech x tanh x dx = – sech x + C
∫cosech x coth x dx = – cosech x + C
Case IV If {(m+1)/n} + P is an integer, we put (a + bxn) = rkxn is the denominator of the
fraction p
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ncerthelp
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ncerthelp
Title: class 12 integration
Description: A basic(but extensive) review of all integrtation concepts for class 12 mathematics ,including formulaes
Description: A basic(but extensive) review of all integrtation concepts for class 12 mathematics ,including formulaes