Integration as the inverse of differentiation (1)

Integration

The reverse process of differentiation

If d/dx[f(x)] = f'(x)

then the integral of f'(x) with respect to x is ∫f'(x) dx = f(x)

Inverse operations

Integration finds the accumulated change to retrieve the original function

Concept of antiderivative

A function F(x) whose derivative is equal to f(x), representing the inverse operation of differentiation

The fundamental theorem of calculus

Integration of a derivative yields the antiderivative

Definition of integration

Finding the cumulative sum or total accumulation of a function over a specified interval

Analytical techniques in integration

Substitution, integration by parts, partial fractions, and trigonometric identities, depending on the complexity of the function

Geometric interpretation in integration

Finding the area under the curve of a function within a given interval

Practical applications of integration

Physics, engineering, economics, and biology, where it is used to calculate quantities such as displacement, work done, volume, and area

Integration formula