Integral

Calculus
definition

Also known as: antiderivative, area under curve

Grade 9-12

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The reverse operation of differentiation; it also computes the exact area under a curve between two points. Integration computes exact areas, volumes, and totals accumulated from known rates of change.

This concept is covered in depth in our integrating rational functions step by step, with worked examples, practice problems, and common mistakes.

Definition

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

💡 Intuition

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

🎯 Core Idea

Integration accumulates; differentiation rates. They're inverses.

Example

\int 2x \, dx = x^2 + C The area under f(x) = 2x from 0 to 3 is 9.

Formula

\int f(x) \, dx = F(x) + C where F'(x) = f(x)

Notation

\int f(x)\,dx denotes the indefinite integral (antiderivative). F(x) is any antiderivative; C is the constant of integration.

🌟 Why It Matters

Integration computes exact areas, volumes, and totals accumulated from known rates of change.

💭 Hint When Stuck

Ask yourself: what function, when differentiated, gives me this integrand? Check by differentiating your answer.

Formal View

F is an antiderivative of f on (a, b) if F'(x) = f(x) for all x \in (a, b). The indefinite integral: \int f(x)\,dx = \{F(x) + C : C \in \mathbb{R}\} where F' = f.

🚧 Common Stuck Point

Always write +C for indefinite integrals—omitting it loses the entire family of antiderivatives.

⚠️ Common Mistakes

  • Forgetting the constant of integration +C on indefinite integrals — without it, you have only one specific antiderivative, not the general solution.
  • Reversing the power rule incorrectly: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, not \frac{x^{n+1}}{n} or \frac{x^n}{n+1}.
  • Thinking the integral of \frac{1}{x} is \frac{x^0}{0} — the power rule doesn't apply when n = -1; the answer is \ln|x| + C.

Frequently Asked Questions

What is Integral in Math?

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

Why is Integral important?

Integration computes exact areas, volumes, and totals accumulated from known rates of change.

What do students usually get wrong about Integral?

Always write +C for indefinite integrals—omitting it loses the entire family of antiderivatives.

What should I learn before Integral?

Before studying Integral, you should understand: derivative.

Prerequisites

derivative

How Integral Connects to Other Ideas

To understand integral, you should first be comfortable with derivative. Once you have a solid grasp of integral, you can move on to definite integral and fundamental theorem.

Want the Full Guide?

This concept is explained step by step in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions →
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Visualization

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Visual representation of Integral