Integral calculator

Type a function of x to get an antiderivative, or tick Definite integral and give the bounds for an exact value. When there is no formula, the full calculator computes the integral numerically, with an error estimate.

Result

−exp(x)+xexp(x)+C

An antiderivative; any constant C works

Steps

Integration by parts, ∫u dv = u·v − ∫v du, choosing u = x (it gets simpler when differentiated) and dv = e^x dx
Standard integral: ∫e^x dx = e^x
So du = 1 dx and v = e^x, giving x·e^x − ∫e^x dx
Standard integral: ∫e^x dx = e^x
Check: differentiating e^x * (x - 1) gives x e^x again
Add the constant of integration: -exp(x) + x exp(x) + C

Graph the function and its antiderivative

How to use it

1
Type the integrand in x, such as x e^x or 1/(x^2 + 1).
2
Press Integrate to get an antiderivative. Add your own constant C.
3
For a definite integral, tick Definite integral and enter the lower and upper bounds.
4
For integrals with no closed form, type int(e^(-x^2), x, 0, 1) in the calculator. It is evaluated by adaptive quadrature with an error estimate.

Standard integrals

Function	Antiderivative
xⁿ (n ≠ −1)	xⁿ⁺¹/(n + 1)
1/x	ln\|x\|
eˣ	eˣ
sin x	−cos x
cos x	sin x
1/(1 + x²)	arctan x
1/√(1 − x²)	arcsin x

Techniques the calculator handles

Integration by parts (∫u dv = uv − ∫v du), substitution, partial fractions for rational functions, and the standard trigonometric and hyperbolic forms are all handled by the algebra system. The example above, x eˣ, is a classic case of integration by parts: with u = x and dv = eˣ dx it gives x eˣ − eˣ.

Some integrals have no elementary antiderivative at all, such as e^(−x²) and sin(x)/x. That is a mathematical fact, not a limitation of the software. Their definite integrals can still be computed to many digits numerically.

Numerical integration with error estimates

In the full calculator, the Analysis tab and the int() function use adaptive Gauss–Kronrod quadrature: each interval is evaluated with a 7-point and a 15-point rule, the difference estimates the error, and intervals are split until the error is below the target. Infinite bounds are handled by a change of variables. The result is shown with its estimated error, so you know how many digits to trust.

Frequently asked questions

How do I calculate a definite integral?

Tick Definite integral and enter the bounds. If an exact value exists it is shown; otherwise use int(f, x, a, b) in the full calculator for a numerical value with an error estimate.

Why is there a + C?

Every function has infinitely many antiderivatives, differing by a constant. C stands for any constant.

Why can’t it integrate e^(−x²)?

That function has no antiderivative made of elementary functions. Its definite integrals can still be computed numerically, and over the whole real line it equals √π.

Can it do integration by parts and substitution?

Yes. Those techniques, partial fractions and the standard forms are all part of the algebra system.