∫ Integral Calculator
Integration is the reverse of differentiation — and the way to find areas under curves. This calculator finds the antiderivative of your function symbolically for all the standard forms (polynomials, trig, exponentials, 1/x, roots, linear substitutions), and if you add bounds it also works out the definite integral.
For functions with no elementary antiderivative, like sin(x²), add bounds and the definite value is still computed numerically to high accuracy.
How to use this calculator
Write the function in terms of x. Bounds are optional — add them when you need the area between two points.
- Type your function f(x) — for example 3*x^2 + sin(x), cos(2*x) or (3*x+1)^2.
- Optionally fill in the lower bound a and upper bound b for a definite integral — numbers or expressions like pi.
- Press Calculate. You get the antiderivative (+ C) for standard forms, and the definite value whenever bounds are given.
- Verify any antiderivative by differentiating it — the Derivative Calculator below does this in one step.
Integral Worked Example
Integrate f(x) = 3x² + sin(x). Term by term: ∫3x² dx = x³, and ∫sin(x) dx = −cos(x), so the antiderivative is x³ − cos(x) + C. Differentiate it back — 3x² + sin(x) — and the original function returns, which proves the answer. For a definite example, ∫₀^π sin(x) dx = 2 exactly: the area under one full arch of the sine wave. Enter bounds 0 and pi and the calculator confirms it.
Frequently asked questions
Which integrals can this calculator solve symbolically?
The standard forms taught in calculus courses: polynomials, x^n for any n, 1/x, e^x, a^x, sin, cos and tan of x or of a linear expression like 2x + 1, roots like sqrt(x), sums and constant multiples of these, and patterns like 1/(1+x²) which gives atan(x). If a form needs integration by parts or a clever substitution, the calculator says so honestly instead of guessing.
What if my integral has no symbolic answer here?
Add lower and upper bounds: the definite integral is then computed numerically to high accuracy (Simpson's rule with 2,000 slices) for any function you can type — even ones like sin(x²) that have no elementary antiderivative at all.
How can I check an antiderivative?
Differentiate it — you must get your original function back. That is also how this calculator's results were verified, and it is the check your teacher expects you to know. Our Derivative Calculator (linked below) will do that differentiation for you.
What does the + C mean?
Every indefinite integral has a whole family of answers differing by a constant, because the derivative of a constant is zero. The + C stands for 'plus any constant' — for definite integrals it cancels out in F(b) − F(a), which is why it only appears on indefinite results.