Integral Calculus PDF Notes, Important Questions And Synopsis
Integral Calculus PDF Notes, Important Questions and Synopsis
- Integration is the inverse process of differentiation. The process of finding the function from its primitive is known as integration or anti-differentiation.
- Finding a function whenever its derivative is given leads to the indefinite form of integrals.
- Finding the area bounded by a graph of a function under certain conditions leads to the definite form of integrals.
- Indefinite and definite integrals together constitute Integral Calculus.
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. , where F(x) is the anti-derivative of f(x).
Functions with the same derivatives differ by a constant.
means 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.
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.
Properties of anti-derivatives:
, for any real number k
where k1, k2, … kn are real numbers and f1, f2, ... fn are real functions.
Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.
Comparison between differentiation and integration:
- Both are operations on functions.
- Both satisfy the property of linearity.
- All functions are not differentiable and all functions are not integrable.
- The derivative of a function is a unique function, but the integral of a function is not unique.
- 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.
- The derivative of a function is defined at a point P and the integral of a function is defined over an interval.
- 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.
- 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.
- Both differentiation and integration are processes involving limits.
- Both are operations on functions.
- Integration can be done using many methods. Prominent among them are
- Integration by substitution
- Integration using partial fractions
- Integration by parts
- Integration using trigonometric identities
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
; substitute x = a sec θ
; substitute x = a sin θ or a cos θ
A function of the form is known as a rational function. Rational functions can be integrated using partial fractions.
Partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function.
Integration using partial fractions:
A rational function can be expressed as the sum of partial fractions. This takes any of the following forms:
- , a ≠ b
- , where cannot be factorised further.
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.
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)]
- , where f1 and f2 are the functions of x.
Definite integral 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’.
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
Methods of evaluating
- Calculate anti-derivative F(x)
- Calculate F(b) – F(a)
A(x) =A(x) = , if x is a point in [a, b].
Fundamental theorems of integral calculus:
- First fundamental theorem: If area function
A(x) = for all x ≥ a and f is continuous on [a, b], then A′(x) = f(x) for all x Î [a, b].
- Second fundamental theorem: If f is a continuous function of x in the closed interval [a, b] and F is an anti-derivative of for all x in domain of f, then
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