What does dx mean in an integral?

In an integral, dx names the variable you add along and stands for a tiny width of x. So ∫ 6x² dx reads as a sum of strips, each 6x² tall and dx wide. Make the strips thinner and thinner, and their total becomes the exact area. Without the dx, the integral would not say which way to slice.

How do you evaluate a definite integral?

You evaluate a definite integral in three moves. Find an antiderivative, a function whose derivative is the one inside the integral. Put in the top limit, then subtract its value at the bottom limit. For 2x from 1 to 3, the antiderivative is x², so 9 − 1 = 8. In this game the bottom limit is always 0, where these antiderivatives are 0.

How do you remember the power rule for integration?

Think of socks and shoes: you put on socks first, but you take off shoes first. Differentiating multiplies by the power and then lowers it by one, so integrating undoes those steps in the opposite order: it adds one to the power, then divides by it. So 12x³ becomes 3x⁴, and differentiating 3x⁴ returns 12x³.

What is the most common mistake in u-substitution?

The most common mistake is dropping the number in front of x inside the bracket. In ∫(3x + 2)⁴ dx, raising the power gives (3x + 2)⁵ over 5, but the bracket grows 3 times as fast as x, so you also divide by 3. The denominator is 3 × 5 = 15, not 5. Differentiating (3x + 2)⁵ / 15 gives (3x + 2)⁴ back exactly.

What does it cost to bring this integral calculus game into a lesson?

Nothing at all: the three levels, from single powers to substitution, open in a browser tab on school laptops or tablets, with no student accounts. For integration practice at the board, ask groups for functions whose area from 0 to 2 is exactly 8. The constant 4, the line 4x and the curve 3x² all work, with three very different shapes. With the schools plan, priced per student, a teacher dashboard tracks who is getting better at integrals.

Math practice · reimagined

Integrals Game

Every area in this integral game starts at x = 0 and stops at 2 or 3, and the substitution level asks only for a denominator. Your rival integrates the same curve, so every step you save lands your fighter’s punch first.

Ages 4–99+ · no download · plays in your browser · 23 languages

Choose your level ▸

What is an integral?

An integral adds up a quantity that keeps changing. For a curve above the x-axis, a definite integral gives the area under it between two values of x, called its limits. Picture a graph of a car’s speed during a trip: the area under that graph is the distance the car covered.

Every level of Integral

Example: ∫(0→2) (3x² + 8x + 5) dx

Integrate one term at a time: 3x² becomes x³, 8x becomes 4x², and the plain 5 becomes 5x.
Put the top limit, x = 2, into x³ + 4x² + 5x: that gives 8 + 16 + 10 = 34.
At the bottom limit, x = 0, every term is 0, so nothing is taken away and you type 34.

Learn it step by step

Integration is differentiation in reverse — it adds up tiny slices to find the AREA under a curve.

The power rule flips: ∫axⁿ dx = a·xⁿ⁺¹/(n+1) — raise the exponent by one, then divide by the new exponent. For ∫2x dx from 0 to 3: that's x² evaluated at 3 → 9.

Integrate, evaluate at the top limit (the bottom is 0 here), and type the area.

Tips and common mistakes

Read the symbols as a sentence: ∫ is a tall S for sum, and dx is a very thin width. So ∫ 6x² dx means adding up thin strips of height 6x².
A common slip is multiplying by the new power instead of dividing. 6x² integrates to 2x³, not 18x³; differentiate your answer, and you should get 6x² back.
On squared paper, draw the line y = x from 0 to 6 and count the squares under it. The triangle covers 18, just as x²/2 at 6 predicts.

How to play

On the first two levels, each round shows an area to find, such as ∫(0→3) 8x³ dx, and you key in the whole number it equals, 162. The substitution level shows ∫(2x+5)³ dx instead and wants only the denominator of its answer: 8. Answer before your rival and Carrot strikes, harder the sooner you are right.

— Quick answers —

Integral: the short FAQ

What does dx mean in an integral?: In an integral, dx names the variable you add along and stands for a tiny width of x. So ∫ 6x² dx reads as a sum of strips, each 6x² tall and dx wide. Make the strips thinner and thinner, and their total becomes the exact area. Without the dx, the integral would not say which way to slice.
How do you evaluate a definite integral?: You evaluate a definite integral in three moves. Find an antiderivative, a function whose derivative is the one inside the integral. Put in the top limit, then subtract its value at the bottom limit. For 2x from 1 to 3, the antiderivative is x², so 9 − 1 = 8. In this game the bottom limit is always 0, where these antiderivatives are 0.
How do you remember the power rule for integration?: Think of socks and shoes: you put on socks first, but you take off shoes first. Differentiating multiplies by the power and then lowers it by one, so integrating undoes those steps in the opposite order: it adds one to the power, then divides by it. So 12x³ becomes 3x⁴, and differentiating 3x⁴ returns 12x³.
What is the most common mistake in u-substitution?: The most common mistake is dropping the number in front of x inside the bracket. In ∫(3x + 2)⁴ dx, raising the power gives (3x + 2)⁵ over 5, but the bracket grows 3 times as fast as x, so you also divide by 3. The denominator is 3 × 5 = 15, not 5. Differentiating (3x + 2)⁵ / 15 gives (3x + 2)⁴ back exactly.
What does it cost to bring this integral calculus game into a lesson?: Nothing at all: the three levels, from single powers to substitution, open in a browser tab on school laptops or tablets, with no student accounts. For integration practice at the board, ask groups for functions whose area from 0 to 2 is exactly 8. The constant 4, the line 4x and the curve 3x² all work, with three very different shapes. With the schools plan, priced per student, a teacher dashboard tracks who is getting better at integrals.

More math to master

Ready to throw down?

Play free ▸