Integral Calculator

What does the integral calculator do?

It does two jobs. For a definite integral it computes the signed area under the curve between the lower and upper limits numerically and reports an error estimate with the result. For an indefinite integral it tries to find the antiderivative symbolically. If one is found it is shown as a formula. If not, the tool says so and does not guess.

How is the definite integral computed?

The method is adaptive Simpson quadrature: the interval is split into panels, Simpson's rule is applied on each panel at two levels of refinement and the panel is halved until the difference drops below the target accuracy. On one panel Simpson's rule reads (b − a)/6 · (f(a) + 4·f((a + b)/2) + f(b)). The result is also computed independently with a 15 point Gauss-Legendre rule and the two are compared. For integrals that are undefined at an end point and for infinite limits the tool switches to the Gauss-Legendre rule, which never evaluates the end points. A limit can be a number, an expression such as pi/2, or inf for infinity.

Which indefinite integrals can it solve?

• Polynomials and powers, 1/x
• sin, cos, tan, exponentials, ln and inverse trigonometric functions with a linear argument: sin(2x + 1)
• A polynomial times an exponential, sine, cosine or ln: x·eˣ, x²·cos x, x·ln x (integration by parts)
• The form f(g(x))·g′(x): 2x·e^(x²), x/(x² + 1), sin x·cos x (substitution)
• Linear substitution: x·√(x + 1), x/(x + 1)
• Quadratic denominators: 1/(x² + 4), 1/√(1 − x²)

How is the result checked?

Every antiderivative found is differentiated and compared with the integrand at many points. A candidate that fails the check is not shown. For a definite integral with a known antiderivative the value F(b) − F(a) is shown next to the numeric result.

Example

Integrate x^2 from 0 to 3. The antiderivative is x³/3. At the upper limit it gives 27/3 = 9 and at the lower limit 0, so the integral is 9. The numeric method returns the same value.

Limits

The symbolic part is not a computer algebra system: it does no partial fractions, no trigonometric substitution and no repeated integration by parts (such as eˣ·sin x), and in those cases it reports that nothing was found. Functions such as e^(−x²) and sin(x)/x have no antiderivative in elementary functions at all. If the integrand is undefined inside the interval (such as 1/x between −1 and 1) the tool gives no result: split the integral at that point and compute two improper integrals. For divergent integrals and very rapidly oscillating integrands the method does not converge and the tool says so. Angles are in radians.