Introduction to Integration:

Integration is one of the two main operations in calculus, the other being differentiation. While differentiation deals with finding the rate of change or slope of a function, integration focuses on finding the original function given its derivative. This is why integration is often referred to as the inverse process of differentiation.



Definition:

If F'(x) = f(x), then the integral of f(x) is given by:

∫ f(x) dx = F(x) + C,

where:

- F(x) is the antiderivative or primitive of f(x).

- C is the constant of integration, which accounts for the fact that differentiation of a constant is zero.

Key Concepts:

1. Indefinite Integral

The indefinite integral represents a family of functions whose derivative is f(x). It is expressed as:

∫ f(x) dx = F(x) + C.

Example: ∫ 2x dx = x² + C, since d/dx(x² + C) = 2x.

2. Constant of Integration:

When integrating, we add C because there are infinitely many antiderivatives that differ by a constant.

Example: ∫ 3 dx = 3x + C.

3. Basic Integration Formulas

4. Properties of Integration

1. Linearity:

  ∫ [af(x) + bg(x)] dx = a∫ f(x) dx + b∫ g(x) dx, where a and b are constants.

2. Constant Multiple Rule:

  ∫ [k ⋅ f(x)] dx = k ∫ f(x) dx, where k is a constant.

3. Addition/Subtraction Rule:

  ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

Integration Techniques:

1. Substitution Method

Used when the integrand contains a composite function.

Steps:

1. Identify a part of the integrand as u and substitute u = g(x).

2. Replace dx with du/g'(x).

3. Integrate with respect to u and revert to x afterward.

Example:

∫ (2x)(x² + 1)³ dx.

Let u = x² + 1, so du = 2x dx.

∫ (2x)(x² + 1)³ dx = ∫ u³ du = u⁴ / 4 + C = (x² + 1)⁴ / 4 + C.

2. By Parts

Used when the integral is a product of two functions.

Formula:

∫ u v dx = u ∫ v dx - ∫ (du/dx ∫ v dx) dx.

Example:

∫ x eˣ dx.

Let u = x and dv = eˣ dx. Then du = dx and v = eˣ.

∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C.

Summary

Integration reverses differentiation and is used to find antiderivatives.

Always include the constant of integration C.

Practice applying basic formulas and techniques like substitution and integration by parts to solve problems.

Practice Questions:

1. Evaluate ∫ (3x² - 2x + 4) dx.

2. Solve ∫ sin(x) cos(x) dx using substitution.

3. Find ∫ x ln(x) dx using integration by parts.

4. Compute ∫ e^(2x)/(1+e^(2x)) dx.