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IIT JEE Maths Integral Calculus

Integral Calculus PDF Notes, Important Questions and Synopsis

SYNOPSIS

  1. Integration is the inverse process of differentiation. The process of finding the function from its primitive is known as integration or anti-differentiation.
  2. Finding a function whenever its derivative is given leads to the indefinite form of integrals.
  3. Finding the area bounded by a graph of a function under certain conditions leads to the definite form of integrals.
  4. Indefinite and definite integrals together constitute Integral Calculus.
  5. If f(x) is a function, then the family of all its anti-derivatives is called the indefinite integral of f(x) with respect to x.
    Indefinite integral, i.e. begin mathsize 12px style integral straight f left parenthesis straight x right parenthesis dx equals straight F left parenthesis straight x right parenthesis plus straight C end style , where F(x) is the anti-derivative of f(x).

  6. Functions with the same derivatives differ by a constant. 

  7. begin mathsize 12px style integral straight f left parenthesis straight x right parenthesis dx end stylemeans an integral of f with respect to x, where f(x) is the integrand, x is the variable of integration and C is the constant of integration.

  8. Geometrically, indefinite integral is the collection of a family of curves, each of which can be obtained by translating one of the curves parallel to itself.

  9. Properties of anti-derivatives:
    begin mathsize 12px style integral left square bracket straight f left parenthesis straight x right parenthesis plus straight g left parenthesis straight x right parenthesis right square bracket dx equals integral straight f left parenthesis straight x right parenthesis dx plus integral straight g left parenthesis straight x right parenthesis dx end style
    begin mathsize 12px style integral kf left parenthesis straight x right parenthesis dx equals straight k integral straight f left parenthesis straight x right parenthesis dx end style, for any real number k
    begin mathsize 12px style integral left square bracket straight k subscript 1 straight f subscript 1 left parenthesis straight x right parenthesis plus straight k subscript 2 straight f subscript 2 left parenthesis straight x right parenthesis plus. ..... plus straight k subscript straight n straight f subscript straight n left parenthesis straight x right parenthesis right square bracket dx equals straight k subscript 1 integral straight f subscript 1 left parenthesis straight x right parenthesis dx plus straight k subscript 2 integral straight f subscript 2 left parenthesis straight x right parenthesis dx plus. ... plus straight k subscript straight n integral straight f subscript straight n left parenthesis straight x right parenthesis dx comma end stylewhere k1, k2, … kn are real numbers and f1, f2, ... fn are real functions.

  10. Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. 

  11. Comparison between differentiation and integration:

    1. Both are operations on functions.
    2. Both satisfy the property of linearity.
    3. All functions are not differentiable and all functions are not integrable.
    4. The derivative of a function is a unique function, but the integral of a function is not unique.
    5. When a polynomial function P is differentiated, the result is a polynomial whose degree is 1 less than the degree of P. When a polynomial function P is integrated, the result is a polynomial whose degree is 1 more than that of P.
    6. The derivative of a function is defined at a point P and the integral of a function is defined over an interval.
    7. Geometrical meaning: The derivative of a function represents the slope of the tangent to the corresponding curve at a point. The indefinite integral of a function represents a family of curves placed parallel to each other having parallel tangents at the points of intersection of the family with the lines perpendicular to the axis.
    8. A derivative is used for determining certain physical quantities such as the velocity of a moving particle when the distance traversed at any time t is known. Similarly, the integral is used in calculating the distance traversed when the velocity at time t is known.
    9. Both differentiation and integration are processes involving limits.

  12. Integration can be done using many methods. Prominent among them are
    1. Integration by substitution
    2. Integration using partial fractions
    3. Integration by parts
    4. Integration using trigonometric identities
  13. A change in the variable of integration often reduces an integral to one of the fundamental integrals. Some standard substitutions are
    x2 + a2 ; substitute x = a tan θ
    begin mathsize 12px style square root of straight x squared minus straight straight a squared end root end style; substitute x = a sec θ
    begin mathsize 12px style square root of straight a to the power of straight 2 minus straight x to the power of straight 2 end root end style; substitute x = a sin θ or a cos θ

  14. A function of the form begin mathsize 12px style fraction numerator straight P left parenthesis straight x right parenthesis over denominator straight Q left parenthesis straight x right parenthesis end fraction end style is known as a rational function. Rational functions can be integrated using partial fractions.

  15. Partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function.

  16. Integration using partial fractions:
    A rational function begin mathsize 12px style fraction numerator straight P left parenthesis straight x right parenthesis over denominator straight Q left parenthesis straight x right parenthesis end fraction end style can be expressed as the sum of partial fractions. This takes any of the following forms:

    1. begin mathsize 12px style fraction numerator px plus straight q over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis end fraction equals fraction numerator straight A over denominator straight x minus straight a end fraction plus fraction numerator straight B over denominator straight x minus straight b end fraction end style , a ≠ b
    2. begin mathsize 12px style fraction numerator px plus straight q over denominator left parenthesis straight x minus straight a right parenthesis squared end fraction equals fraction numerator straight A over denominator straight x minus straight a end fraction plus straight B over left parenthesis straight x minus straight a right parenthesis squared end style
    3. begin mathsize 12px style fraction numerator px squared plus qx plus straight r over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis left parenthesis straight x minus straight c right parenthesis end fraction equals fraction numerator straight A over denominator straight x minus straight a end fraction plus fraction numerator straight B over denominator straight x minus straight b end fraction plus fraction numerator straight C over denominator straight x minus straight c end fraction end style
    4. begin mathsize 12px style fraction numerator px squared plus qx plus straight r over denominator left parenthesis straight x minus straight a right parenthesis squared left parenthesis straight x minus straight b right parenthesis end fraction equals fraction numerator straight A over denominator straight x minus straight a end fraction plus straight B over left parenthesis straight x minus straight a right parenthesis squared plus fraction numerator straight C over denominator straight x minus straight b end fraction end style
    5. begin mathsize 12px style fraction numerator px squared plus qx plus straight r over denominator left parenthesis straight x minus straight a right parenthesis left parenthesis straight x squared plus bx plus straight c right parenthesis end fraction equals fraction numerator straight A over denominator straight x minus straight a end fraction plus fraction numerator Bx plus straight C over denominator straight x squared plus bx plus straight c end fraction end style, where begin mathsize 12px style straight x squared plus bx plus straight c end style  cannot be factorised further.
  17. To find the integral of the product of two functions, integration by parts is used. 

    I and II functions are chosen using the ILATE rule:
    I - Inverse trigonometric
    L - Logarithmic
    A - Algebraic
    T - Trigonometric
    E - Exponential
    is used to identify the first function.

  18. Integration by parts:
    Integral of the product of two functions = (first function) × (integral of the second function) – integral of [(differential coefficient of the first function) × (integral of the second function)]

    1. begin mathsize 12px style integral straight f subscript 1 left parenthesis straight x right parenthesis. straight f subscript 2 left parenthesis straight x right parenthesis dx equals straight f subscript 1 left parenthesis straight x right parenthesis integral straight f subscript 2 left parenthesis straight x right parenthesis dx minus integral open square brackets straight d over dx straight f subscript 1 left parenthesis straight x right parenthesis. integral straight f subscript 2 left parenthesis straight x right parenthesis dx close square brackets dx end style , where f1 and f2 are the functions of x. 
    2. begin mathsize 12px style integral straight e to the power of straight x open parentheses straight f open parentheses straight x close parentheses plus straight f apostrophe open parentheses straight x close parentheses close parentheses dx  =  straight e to the power of straight x straight f left parenthesis straight x )+ straight C end style

  19. Definite integral begin mathsize 12px style integral from straight a to straight b of straight f left parenthesis straight x right parenthesis dx end style of the function f(x) from limits a to b represents the area enclosed by the graph of the function f(x), the x-axis and the vertical markers x = ‘a’ and x = ‘b’.

  20. Definite integral as the limit of a sum: The process of evaluating a definite integral by using the definition is called integration as the limit of a sum or integration from first principles.
    For any function f(x), it is given as

    begin mathsize 12px style integral from straight a to straight b of straight f left parenthesis straight x right parenthesis dx equals left parenthesis straight b minus straight a right parenthesis limit as straight n rightwards arrow straight infinity of 1 over straight n open square brackets straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus. ... plus straight f left parenthesis straight a plus left parenthesis straight n minus 1 right parenthesis straight h close square brackets text    end text where text    end text straight h equals fraction numerator straight b minus straight a over denominator straight n end fraction end style

  21. Methods of evaluating begin mathsize 12px style integral from straight a to straight b of straight f left parenthesis straight x right parenthesis dx end style

    1. Calculate anti-derivative F(x)
    2. Calculate F(b) – F(a)
  22. Area function
    A(x) =A(x) =begin mathsize 12px style integral from straight a to straight x of straight f left parenthesis straight x right parenthesis dx end style , if x is a point in [a, b].

  23. Fundamental theorems of integral calculus:

  1. First fundamental theorem: If area function

    A(x) = begin mathsize 12px style integral from straight a to straight x of straight f left parenthesis straight x right parenthesis dx end style  for all x ≥ a and f is continuous on [a, b], then A′(x) = f(x) for all x Î [a, b].
  2. Second fundamental theorem: If f is a continuous function of x in the closed interval [a, b] and F is an anti-derivativebegin mathsize 12px style straight d over dx straight F left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis end style of   for all x in domain of f, then
     begin mathsize 12px style integral from straight a to straight b of straight f left parenthesis straight x right parenthesis dx equals open square brackets straight F left parenthesis straight x right parenthesis plus straight C close square brackets subscript straight a superscript straight b equals straight F left parenthesis straight b right parenthesis minus straight F left parenthesis straight a right parenthesis end style.