Detailed Notes on Integrals

7.1 Introduction

Integral calculus focuses on the concept of the integral—the process of finding an antiderivative or calculating the area under curves. It complements differential calculus, which deals with derivatives and slope calculations.

Concept of Antiderivatives

Given a differentiable function f on an interval I, if we know its derivative f᾽, we may want to find the original function f, which is an antiderivative or primitive of f᾽. The set of all antiderivatives of f᾽ is represented by the indefinite integral:

[ \int f'(x) , dx = f(x) + C ]

where C is a constant.

Definite Integrals

The process leads to definite integrals, which help in determining the area under a curve, from point a to point b, expressed as:

[ \int_a^b f(x) , dx = F(b) - F(a) ]

where F is any antiderivative of f.

7.2 Integration as an Inverse Process of Differentiation

Integration serves as the inverse operation to differentiation. For instance, knowing derivatives such as:

• ( \frac{d}{dx} (sin x) = cos x \
• \frac{d}{dx} (x^2) = 2x \
• \frac{d}{dx} (e^x) = e^x \

The corresponding integrals are:

• ( \int cos x , dx = sin x + C \
• \int 2x , dx = x^2 + C \
• \int e^x , dx = e^x + C \

Properties of Indefinite Integrals

1. Linearity: ( \int [f(x) + g(x)] , dx = \int f(x) , dx + \int g(x) , dx \
2. Constant factor: ( \int k f(x) , dx = k \int f(x) , dx \

7.3 Methods of Integration

7.3.1 Integration by Substitution

Substitution aids in simplifying integrals. For instance, if we have: [ I = \int f(g(x)) g'(x) , dx = \int f(t) , dt ] where we make the substitution ( t = g(x) ).

7.3.2 Integration Using Partial Fractions

This method applies to rational functions, where the integrand can be simplified into simpler fractions.

7.3.3 Integration by Parts

This technique derives from the product rule of derivatives, expressed as follows: [ \int u ,, dv = uv - \int v , du ] Choosing which function to differentiate (u) and which to integrate (v) effectively is vital.

7.4 Integrals of Some Particular Functions

Several standard integrals emerge repeatedly, including:

• ( \int \frac{1}{x} , dx = log |x| + C \
• \int e^x , dx = e^x + C \
• \int sin x , dx = -cos x + C \

7.5 Integration by Parts

Key examples illustrate the effectiveness of integrating by parts, like integrating polynomial and trigonometric functions together.

7.6 Evaluation of Definite Integrals

Definite integrals yield specific values representing the area under a curve: [ \int_a^b f(x) , dx = F(b) - F(a) ] where F is an antiderivative of f.

7.7 Further Properties of Definite Integrals

• Adjusting limits can simplify definite integrals.
• If the function is odd, the integral over symmetric limits results in zero.

Key Theorems

Fundamental Theorem of Calculus

1. If f is continuous on [a, b], then A'(x) = f(x) and provides a connection between the derivative of an integral and the original function.
2. The second part establishes that if F is an antiderivative of f then the evaluation of the definite integral leads to F(b) - F(a).

Key Points to Remember

1. Integration is the inverse of differentiation.
2. Indefinite integrals are expressed as ( \int f(x) ) and include a constant of integration (C).
3. Definite integrals provide the area under a curve between two points a and b.
4. Integration Techniques include substitution, partial fractions, and integration by parts.
5. Properties of integrals include linearity and constant factors that simplify calculations.
6. The Fundamental Theorem of Calculus links indefinite and definite integrals with derivatives.
7. Special standard integrals frequently encountered have specific formulae for quick reference.
8. Antiderivatives can often be found using algebraic techniques, relying on known derivatives of basic functions.

1. Integration is inverse to differentiation.
2. Indefinite integrals include an arbitrary constant (C).
3. The area under a curve is calculated using definite integrals.
4. Integration techniques include substitution, partial fractions & integration by parts.
5. Integrals have important properties like linearity.
6. The Fundamental Theorem of Calculus connects integration and differentiation.
7. Use standard integrals for common functions for efficient computation.
8. Antiderivatives may involve various algebraic methods.