Home

Indefinite Integrals Online Calculator

Indefinite-integral explorer. Enter f(x) and a base point a to accumulate ∫_a^x f(t)dt, visualizing the antiderivative and constant C.

Show more ▾
∫ f(x) dx = F(x) + C
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Indefinite Integrals

1. Definition
If F′(x)=f(x) on an interval I, F(x) is an antiderivative of f(x). The family is written ∫ f(x) dx = F(x) + C, where C is the arbitrary constant of integration.
2. Symbols
SymbolsMeaning
∫Integral sign
f(x)Integrand
F(x)One antiderivative of f(x)
CConstant of integration
aBase point of accumulation
bEvaluation point x
3. How it works
  • When no closed form exists, numerically accumulate ∫_a^b f(t)dt to get one definite antiderivative F(b)−F(a);
  • Changing the base a only shifts the result by a constant C;
  • The indefinite integral is therefore a family of parallel functions, not a single one.
4. Steps
  1. Enter f(x), e.g. x^2;
  2. Enter a base point a (e.g. 0) and a point b (e.g. 1);
  3. Click Calculate to get F(b)−F(a);
  4. Change a and recompute to see only a vertical shift.
5. Example
Example: f(x)=x². An antiderivative is F(x)=x³/3. With a=0,b=1: F(1)−F(0)=1/3≈0.3333. The tool gives 0.333333.
6. Pitfalls
Always append +C to an indefinite integral, or you name just one antiderivative;
Antiderivatives differ by a constant; any two share the same shape;
The numeric value depends on a; fix a before comparing.

❓ FAQ | Indefinite Integrals

Why must I add +C?
Antiderivatives are not unique: if F′=f then (F+C)′=f for any constant C, so the result is a whole family.
What does the base point a do?
It only chooses which member of the family: F(b)−F(a). Changing a just changes C; the shape is unchanged.
Can it give a closed-form antiderivative?
No. It uses adaptive Simpson integration to give one antiderivative value at b, building the intuition that accumulation is an antiderivative.
Which functions are supported?
+ - * / power ^ parentheses, plus sin cos tan sec cot sqrt exp log abs asin atan and constants pi, e. Never omit *.
Why is there a tiny gap vs. hand calculation?
Numerical integration has truncation and rounding error, typically around 1e-6, which does not change the result.