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.