Integral Calculus
Integration: Process of finding antiderivatives or integrals of functions. Denoted by ∫ (integral symbol).
Indefinite Integral: Represents a family of functions whose derivative is the integrand. Notation: ∫ f(x) dx + C (where C is the constant of integration).
Fundamental Concepts:
1. Definite Integral:
 Represents the accumulation of quantities over an interval.
 Notation: ∫ (integral symbol)
 ∫[a, b] f(x) dx denotes the definite integral of f(x) from 'a' to 'b'.
2. Antiderivative:
 The reverse of differentiation.
 Finding functions whose derivatives correspond to a given function.
 Notation: ∫ f(x) dx = F(x) + C, where F(x) is the antiderivative of f(x), and 'C' is the constant of integration.
3. Fundamental Theorem of Calculus:
 Connects differentiation and integration.
 Relates definite integrals to antiderivatives.
 States that if 'f' is continuous on [a, b], then ∫[a, b] f(x) dx = F(b)  F(a), where F is an antiderivative of f.
4. Integration Techniques:

Substitution Method:
 Involves substituting a new variable to simplify the integral.
 Useful for complex functions or when using trigonometric identities.

Integration by Parts:
 Relates the integral of a product of two functions to the product of their antiderivatives.
 Involves choosing which function to differentiate and which to integrate.

Partial Fractions:
 Used for integrating rational functions by breaking them down into simpler fractions.
 Helpful when dealing with complex fractions.

Trigonometric Integrals:
 Utilizes trigonometric identities to simplify integrals involving trigonometric functions.
Applications:
1. Area Under a Curve:
 Calculating the area between the curve of a function and the xaxis.
2. Volume of Revolution:
 Finding the volume generated by rotating a region about an axis using integrals.
3. Arc Length:
 Determining the length of a curve using integrals.
4. Surface Area:
 Calculating the surface area of a threedimensional object using integrals.
Tips for Problem Solving:
 Identify the Type of Integral: Determine which technique might be best suited for the given integral.
 Simplify and Manipulate: Simplify the integrand or manipulate the equation to make integration more manageable.
 Use Properties of Integration: Utilize properties such as linearity and the rules of integration to simplify the problem.
 Check for Mistakes: Integration often involves multiple steps, so check each step carefully for errors.