Calculus · Step-by-step guide

What Is an Integral?

Understand definite and indefinite integrals as accumulation, signed area, and reverse differentiation.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

An indefinite integral is a family of antiderivatives; a definite integral measures net accumulation over an interval. The Fundamental Theorem of Calculus connects them: ∫abf(x)dx=F(b)−F(a)\displaystyle \int_a^b f(x)dx=F(b)-F(a) when F′ = f.

∫abf(x) dx=F(b)−F(a)\displaystyle \int_a^b f(x)\,dx=F(b)-F(a)
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Accumulation from small pieces

A definite integral is the limit of sums of thin rectangles or other small contributions.

2

Signed area

Area above the x-axis contributes positively and area below contributes negatively.

3

Antiderivatives reverse derivatives

Indefinite integrals include +C because many functions share the same derivative.

02

See the structure

A picture makes the relationships easier to remember.

ab
The shaded region represents accumulated signed area between the curve and the x-axis.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Basic antiderivative

Find ∫3x2dx\displaystyle \int 3x^2dx.

  1. Reverse the power rule.
  2. Increase the exponent to 3 and divide by 3.
  3. Include the constant of integration.
Answerx3+C\displaystyle x^3+C
Example 2Definite integral

Evaluate ∫02x dx\displaystyle \int_0^2 x\,dx.

  1. An antiderivative is x2/2\displaystyle x^2/2.
  2. Evaluate upper minus lower.
  3. 22/2−0=2\displaystyle 2^2/2-0=2.
Answer2
Example 3Net area

A function has area 5 above the axis and area 2 below.

  1. Above contributes +5.
  2. Below contributes −2.
  3. Add signed contributions.
AnswerIntegral = 3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Forgetting +C on an indefinite integral.

  • Treating net area as always positive.

  • Evaluating lower minus upper.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Basic Integration

Practice antiderivatives and definite integrals.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find ∫3x2 dx\displaystyle \int 3x^2\,dx.

Show hint

Reverse the power rule.

Show worked solution
  1. Increase the exponent from 2 to 3.
  2. Divide by the new exponent: 3x3/3\displaystyle 3x^3/3.
  3. Add the constant of integration.

Answer: x3+C\displaystyle x^3+C

2

Evaluate ∫02x dx\displaystyle \int_0^2 x\,dx.

Show hint

Use an antiderivative x2/2\displaystyle x^2/2.

Show worked solution
  1. [x2/2]02\displaystyle [x^2/2]_0^2.
  2. 22/2−0=2\displaystyle 2^2/2-0=2.

Answer: 2\displaystyle 2

3

Evaluate ∫134 dx\displaystyle \int_1^3 4\,dx.

Show hint

A constant integral is rectangle area.

Show worked solution
  1. Antiderivative: 4x\displaystyle 4x.
  2. 4(3)−4(1)=8\displaystyle 4(3)-4(1)=8.

Answer: 8\displaystyle 8

4

Find ∫(2x+5) dx\displaystyle \int (2x+5)\,dx.

Show hint

Integrate term by term.

Show worked solution
  1. ∫2x dx=x2\displaystyle \int2x\,dx=x^2.
  2. ∫5 dx=5x\displaystyle \int5\,dx=5x.
  3. Add C.

Answer: x2+5x+C\displaystyle x^2+5x+C

5

If F′(x)=f(x)\displaystyle F'(x)=f(x), simplify ∫27f(x) dx\displaystyle \int_2^7 f(x)\,dx.

Show hint

Use the Fundamental Theorem of Calculus.

Show worked solution
  1. Evaluate the antiderivative at the upper bound.
  2. Subtract its value at the lower bound.

Answer: F(7)−F(2)\displaystyle F(7)-F(2)

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

An indefinite integral is a family of antiderivatives; a definite integral measures net accumulation over an interval. The Fundamental Theorem of Calculus connects them: ∫abf(x)dx=F(b)−F(a)\displaystyle \int_a^b f(x)dx=F(b)-F(a) when F′ = f.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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