Free lesson · Integral calculus

Core integral rules: undo a derivative, then check

Open interactive lesson Practice calculations Explore labs

Start with the idea

An antiderivative reverses differentiation.
Check your answer by differentiating it.

Symbols, units & horizon

x: dimensionless input
n: nonnegative integer exponent for the rule stated here
c: constant integrand value
C: arbitrary integration constant
dx: integrate with respect to x
∫: indefinite integration, producing a family of antiderivatives
F: an antiderivative

When and why to use this

Use an antiderivative to reconstruct accumulation from a rate. Definite integrals then compare endpoint values; substitution later reverses the chain rule.

Constant rule: $\int c d x = c x + C$ .
Power rule: $\int x^{n} d x = \frac{x^{n + 1}}{(n + 1)} + C$ for nonnegative integer n here.
Increase the power by one. Divide by that new power.
Sum rule: integrate each term and add the results.
A fixed multiplier stays outside the integral.
The arbitrary constant C has derivative zero. Different C values give the same derivative.

Changing C shifts an antiderivative vertically. It does not change any of its slopes.
For F(x)=x²+3x+C, choosing C=5 gives F(0)=5 and F(2)=15. Choosing C=0 gives 0 and 10.
The endpoint difference is 10 in both cases: the same constant is added to both endpoints and cancels when you subtract.

\int (2 x + 3) d x = x^{2} + 3 x + C

Core rule · definition and worked arithmetic

Core integral rules: undo a derivative, then check

Differentiate $\frac{x^{n + 1}}{(n + 1)}$ : the power rule gives $\frac{(n + 1) x^{n}}{(n + 1)} = x^{n}$ .
Split the example into $\int 2 x d x + \int 3 d x$ .
The terms integrate to $x^{2}$ and $3 x$ . Add one arbitrary constant C.
Check: $\frac{d}{d x} (x^{2} + 3 x + C) = 2 x + 3$ .

Work it by hand

Choose C=0. Then F(2)=2²+3×2=10 and F(0)=0. The next lesson explains why F(2)−F(0)=10 gives the definite integral on [0,2].

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Further reading: OpenStax · Definite integrals · textbook checked 12 September 2026 ↗

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def antiderivative(x, constant=0):
    return x*x+3*x+constant

def check_derivative(x):
    return 2*x+3

print(antiderivative(2), check_derivative(2))

Continue learning

Integral Calculus: Accumulation & Area — all lessons

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations