Integral Calculus

7.1 Introduction

  • Differential Calculus focuses on the concept of the derivative, which originated from the need to define tangent lines and calculate their slopes for function graphs.

  • Integral Calculus is concerned with determining areas under function graphs.

Motivation for Integration

  • Given a differentiable function f over an interval I , we can pose the question: if we know the derivative f' at every point in I , can we recover the function f ?

  • The potential functions whose derivative yields f are referred to as anti-derivatives or primitives of f .

  • The mathematical representation for all anti-derivatives of a function is the indefinite integral:

    • Integration: The process of finding anti-derivatives.

Integration is used to find original function from derivatives

Types of Problems in Integral Calculus

  1. Finding a function when its derivative is known.

  2. Calculating area bounded by the graph of a function under certain constraints.

1. Motion related prob.s(Kinematics), Civil Engg.(Structure Based)
2.Applications in economics (e.g., finding consumer and producer surplus), physics (e.g., work done by a variable force), and probability (e.g., continuous random variables). Can only be done using definite int

Forms of Integrals

  • From these problems arise two forms of integrals:

    • Indefinite integrals, which denote a family of functions.

    • Definite integrals, which yield numerical values associated with areas.

7.2 Integration as an Inverse Process of Differentiation

  • Integration serves as the reverse operation to differentiation. Rather than differentiating a function, we can instead be given a derivative and asked to derive the primitive (original function).

Examples of Anti-derivatives

  • Fundamental anti-derivatives:

    1. \frac{d}{dx}(\sin x) = \cos x implies \sin x is an anti-derivative of \cos x .

    2. \frac{d}{dx} x^3 = 3x^2 implies x^3 is an anti-derivative of 3x^2 .

    3. \frac{d}{dx}(e^x) = e^x implies e^x is an anti-derivative of itself.

Non-uniqueness of Anti-derivatives

  • For any real constant C , the derivative of the function including C will yield the same result. The anti-derivatives are essentially unique up to addition of a constant.

    • Example: \text{Anti-derivatives of } \cos x are \sin x + C for any real constant C .

Notation for Indefinite Integrals

  • Indefinite integral notation:

    • \text{If } f(x) = F'(x), \text{ then } \text{the indefinite integral is: } \frac{f(x)}{dx} = \text{F}(x) + C .

  • Integral notation is given as follows:

    • \text{where } \frac{dy}{dx} = f(x) \text{ becomes } y = \text{F}(x) + C .

Properties of Indefinite Integrals

  • Integration respects certain properties:

    1. \int \frac{d}{dx} f(x) dx = f(x) .

    2. If f and g share the same derivatives, then they differ by a constant.

Standard Integral Formulas

  • These include derivatives and integrals for common functions, for instance:

    • \frac{d}{dx} x^n = nx^{n-1} \text{ leads to } \text{Integral of } x^n = \frac{x^{n+1}}{n+1} + C, n \neq -1 \text{.} .

    • \frac{d}{dx}(\sin x) = \cos x \text{ implies } \text{Integral of } \cos x = \sin x + C \text{.} .

7.2.1 Properties of Indefinite Integrals

Key Properties

  1. Integration and Differentiation Are Inverses:

    • \frac{d}{dx} \bigg( \text{F}(x) + C \bigg) = f(x) \text{.} .

  2. Equivalence Class of Anti-derivatives:

    • Two indefinite integrals differing only by a constant describe the same family of curves.

  3. Linearity:

    • \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \text{ means integration respects values and constants.}

  4. Scaling:

    • For real number k : k \frac{d}{dx} f(x) = \frac{d}{dx}(kf(x)) \text{.}

  5. Generalized Linear Properties.

Evaluation of Anti-derivatives by Inspection

  • To find anti-derivatives based intuitively on known derivatives, herein are examples provided:

    1. \frac{d}{dx}(\sin 2x) = 2 \cos 2x gives \frac{1}{2} \sin 2x \text{.}

    2. 3x^2 + 4x^3 yields x^3 + x^4 + C \text{.} .

Additional Examples

  1. Integrate 3x - 1 :

    • (\frac{3x^2}{2} - x + C)\text{.}.

  2. Integrate 2 \cos x :

    • (2 \sin x + C)\text{.}.

Conclusion

  • Integral calculus and its operations through integration facilitate solving for areas and anti-derivatives, advancing our ability to manipulate functions effectively.