Definite Integral Solver

Q: What's the difference between a definite and indefinite integral?

An indefinite integral returns a family of antiderivatives F(x) + C. A definite integral evaluates that antiderivative between two limits a and b, giving a single number: F(b) − F(a).

Q: What does a definite integral represent geometrically?

It's the signed area between the function's graph and the x-axis over the interval [a, b]. Areas above the axis count positive, areas below count negative.

Q: Does the solver handle improper integrals?

Yes — when one or both limits are infinite, or when the integrand has a singularity in the interval. It evaluates the appropriate limit and tells you whether the integral converges.

Q: Why does my integral return 'undefined' or 'divergent'?

The integral doesn't converge. Common causes: a non-integrable singularity (like 1/x at 0), or oscillation that doesn't settle (like sin(x) on [0, ∞)).

Q: Can I use it for area between two curves?

Yes. Compute ∫(top − bottom) dx over the interval where they intersect. The solver returns the signed difference, so order matters.

Q: Does it work with parametric or polar integrals?

Yes for polar (using A = ½∫r² dθ for area). For parametric area you'd compute ∫y(t)·x'(t) dt — set up the integrand explicitly.

Q: How does the solver compute the answer?

When a closed-form antiderivative exists, it applies the Fundamental Theorem of Calculus. When it doesn't (like ∫e^(-x²) dx), it uses adaptive numerical quadrature with controlled error tolerance.

Q: How precise are the numerical answers?

Typically accurate to 8–10 significant figures for well-behaved integrands. Highly oscillatory or near-singular cases drop to 4–6 figures — the solver flags those when it detects them.

Solve the definite integral and visualize the area under the curve.

Function f(x)

Lower bound (a)

Upper bound (b)

Use this free definite integral solver & calculator to compute exact and numerical values of integrals. Enter your function and bounds to get the result with step-by-step antiderivative computation. Perfect for calculus students and anyone needing to calculate areas under curves.

What is a Definite Integral?

A definite integral calculates the net signed area between a function and the x-axis over a specific interval. Unlike indefinite integrals which give you a family of functions, definite integrals produce a single numerical value.

The notation $ \int_a^b f(x)\,dx $ represents the integral of $ f(x) $ from $ a $ to $ b $, where $ a $ is the lower limit and $ b $ is the upper limit of integration.

The Fundamental Theorem of Calculus

If $ F(x) $ is an antiderivative of $ f(x) $, then:

$$\int_a^b f(x)\,dx = F(b) – F(a)$$

This powerful theorem connects differentiation and integration, showing they’re inverse operations.

Signed Area

Above the x-axis

When $ f(x) > 0 $, the area contribution is positive. The region between the curve and the x-axis adds to the total.

Below the x-axis

When $ f(x) < 0 $, the area contribution is negative. This is why we call it “net signed area” rather than just area.

Properties of Definite Integrals

Additivity

$$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$$

You can split integrals at any point.

Reversal

$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$

Swapping limits changes the sign.

Linearity

$$\int_a^b [af(x) + bg(x)]\,dx = a\int_a^b f(x)\,dx + b\int_a^b g(x)\,dx$$

Constants factor out, and integrals distribute over addition.

Applications

Physics

Work done by a force
Displacement from velocity
Charge from current
Energy calculations

Geometry

Area between curves
Volume of solids
Arc length
Surface area

Probability

Expected values
Cumulative distribution functions
Probability density integration

Common Integrals

Function	Integral
$ x^n $	$ \frac{x^{n+1}}{n+1} $
$ \frac{1}{x} $	\( \ln
$ e^x $	$ e^x $
$ \sin(x) $	$ -\cos(x) $
$ \cos(x) $	$ \sin(x) $

See more Integration Formulas.

Frequently Asked Questions

What’s the difference between a definite and indefinite integral?

An indefinite integral returns a family of antiderivatives F(x) + C. A definite integral evaluates that antiderivative between two limits a and b, giving a single number: F(b) − F(a).

What does a definite integral represent geometrically?

It’s the signed area between the function’s graph and the x-axis over the interval [a, b]. Areas above the axis count positive, areas below count negative.

Does the solver handle improper integrals?

Yes — when one or both limits are infinite, or when the integrand has a singularity in the interval. It evaluates the appropriate limit and tells you whether the integral converges.

Why does my integral return ‘undefined’ or ‘divergent’?

The integral doesn’t converge. Common causes: a non-integrable singularity (like 1/x at 0), or oscillation that doesn’t settle (like sin(x) on [0, ∞)).

Can I use it for area between two curves?

Yes. Compute ∫(top − bottom) dx over the interval where they intersect. The solver returns the signed difference, so order matters.

Does it work with parametric or polar integrals?

Yes for polar (using A = ½∫r² dθ for area). For parametric area you’d compute ∫y(t)·x'(t) dt — set up the integrand explicitly.

How does the solver compute the answer?

When a closed-form antiderivative exists, it applies the Fundamental Theorem of Calculus. When it doesn’t (like ∫e^(-x²) dx), it uses adaptive numerical quadrature with controlled error tolerance.

How precise are the numerical answers?

Typically accurate to 8–10 significant figures for well-behaved integrands. Highly oscillatory or near-singular cases drop to 4–6 figures — the solver flags those when it detects them.