Integral Calculator

What is this?

An Integral Calculator finds the area under a curve. An indefinite integral returns the antiderivative - the function whose derivative is the one you typed - plus a constant of integration. A definite integral returns a number: the signed area between two limits.

How to Use the Integral Calculator

An integral calculator finds the area under a curve. There are two kinds. An indefinite integral asks for the antiderivative - the function whose derivative is the one you typed. A definite integral asks for a number: the signed area between two limits.

This tool does both, and treats them differently on purpose. Antiderivatives are found symbolically and then verified by differentiating them back. Definite integrals are computed numerically, which works even for functions that have no antiderivative you can write down.

1. Type the Function

Use x as the variable, for multiplication and ^ for powers. So 3x squared plus sine x is typed as 3x^2 + sin(x). Constants pi and e are recognised.

2. Choose Indefinite or Definite

The Indefinite tab returns the antiderivative plus a constant of integration. The Definite tab asks for limits and returns a number.

3. Enter the Limits

For a definite integral, enter the lower and upper limits. They can be numbers or expressions using pi and e, so pi/2 is fine.

4. Read the Result

An antiderivative comes with the accuracy of its derivative check. A definite integral comes with an error estimate and, when a closed form exists, a comparison against it.

Key Formulas Used in the Calculator

The Power Rule

$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C, \qquad n\neq -1$

The exception matters: at n = −1, the formula divides by zero, and the answer is the natural logarithm instead. ∫ 1/x dx = ln|x| + C.

Linearity

$\int \left(a f + b g\right)\,dx = a\int f\,dx + b\int g\,dx$

Constants come outside the integral and sums split apart, which is what lets a polynomial be integrated term by term.

Linear Inner Argument

$\int f(ax+b)\,dx = \frac{F(ax+b)}{a}+C$

The chain rule in reverse. Since the derivative of ax + b is the constant a, integrating sin(3x) gives −cos(3x)/3 rather than −cos(3x).

The Fundamental Theorem

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

If you have an antiderivative F, a definite integral is just the difference of its values at the two limits. This calculator uses it as an independent check on the numeric answer.

Benefits

• Every antiderivative is verified by differentiating it back before being shown

• Says no closed form was found rather than returning a wrong answer

• Definite integrals work even without an elementary antiderivative

• Adaptive quadrature concentrates effort where the function changes fastest

• Cross-checks the numeric answer against the closed form when one exists

• Reports an error estimate rather than implying false precision

• Accepts pi and e in the limits

When & Where to Use

• Checking an antiderivative you worked out by hand

• Finding the area under a curve between two points

• Computing a definite integral that has no elementary antiderivative

• Verifying homework before submitting it

• Working out displacement from a velocity function

• Finding the mean value of a function over an interval

• Confirming a result from an integral table

Who Should Use This Calculator

The Integral Calculator is for calculus students checking their work, engineers and scientists needing a definite integral quickly, and anyone who wants a numeric answer for an integral that cannot be done in closed form.

Tips to Get the Best Deal

Write multiplication explicitly: 3*x, not 3x

Remember the constant of integration on every indefinite integral

A definite integral is signed - area below the axis counts as negative

If no closed form is found, the definite tab will still give a number

Check for asymptotes between your limits; the integral may not converge

Reversing the limits negates the answer

Differentiating your answer is the quickest way to check any integral

Pro Tips

• Write multiplication explicitly: 3*x, not 3x.

• Remember the constant of integration on every indefinite integral.

• A definite integral is signed - area below the axis counts as negative.

• If no closed form is found, the definite tab still gives a number.

• Check for asymptotes between your limits; the integral may not converge.

• Differentiating your answer is the quickest way to check any integral.