Math foundations · 7–10 min

Integration basics: work backwards from a derivative.

Integration is closely connected to differentiation. A derivative describes how a function changes; an antiderivative is a function whose derivative gives you the expression you started with. Learning a few rules first makes many beginner integrals manageable, while harder expressions may require substitution, integration by parts or other techniques.

1. Think of an antiderivative

Because the derivative of x³ is 3x², an antiderivative of 3x² is x³. In an indefinite integral, we write the family of antiderivatives using the integral symbol and include a constant.

Example: ∫ 3x² dx = x³ + C

The symbol dx indicates integration with respect to x, and C represents any constant. Differentiating x³ + 5 or x³ − 12 gives 3x², because the derivative of a constant is zero.

2. The power rule for integration

For a power xⁿ, add one to the exponent and divide by the new exponent. This rule applies when n ≠ −1.

Power rule

∫ xⁿ dx = x⁽ⁿ⁺¹⁾/(n + 1) + C, for n ≠ −1

Increase the exponent by one. For x², the new exponent is 3.
Divide by the new exponent. The result is x³/3.
Add +C. The indefinite integral is x³/3 + C.

Worked example A

∫ x² dx = x³/3 + C. Check it by differentiating: the derivative of x³/3 is x².

Worked example B

∫ 4x dx = 4 · (x²/2) + C = 2x² + C. Differentiate 2x² + C and you get 4x.

Worked example C: a constant

∫ 5 dx = 5x + C, because the derivative of 5x is 5. A constant number can be treated as a coefficient multiplied by x⁰, though the direct rule is even simpler to remember.

3. Add and subtract terms separately

For sums and differences of basic terms, integrate each term on its own, keeping its coefficient. For example:

∫ (3x² + 2x − 4) dx = x³ + x² − 4x + C

Check by differentiating: 3x² + 2x − 4 is recovered. Notice that the constant of integration is written once at the end.

4. What changes in a definite integral?

A definite integral has lower and upper limits, such as ∫ from a to b f(x) dx. Find an antiderivative F(x), then calculate F(b) − F(a). The constant C is not needed in the final calculation because it cancels when the two values are subtracted.

Example: ∫ from 0 to 2 2x dx = [x²] from 0 to 2 = 2² − 0² = 4

Geometrically, a definite integral represents signed area: portions below the horizontal axis count as negative. It is not always the same as total area unless the function remains nonnegative or you account for the sign changes.

5. Frequent mistakes to check

Forgetting +C: include it for an indefinite integral.
Adding one but not dividing: ∫ x² dx is x³/3 + C, not simply x³ + C.
Dividing by the old exponent: divide by the new exponent after increasing it.
Using the power rule for 1/x: the case n = −1 is an exception; ∫ 1/x dx = ln|x| + C.
Skipping the check: differentiate your result. If it does not reproduce the original integrand, find the error.

6. Practice from memory

Try each integral before opening the answer. Write the steps and include the constant when the integral is indefinite.

∫ x³ dx
∫ 6x² dx
∫ (2x + 3) dx
∫ from 1 to 3 2x dx

Show solutions

x⁴/4 + C
2x³ + C
x² + 3x + C
[x²] from 1 to 3 = 9 − 1 = 8

After checking the results, close the solutions and redo any problem you missed. A correct answer reached by copying is not as useful as one you can reproduce independently.

Use active recall to practise recalling rules and spaced repetition to schedule another attempt later.