DE-Formula Integration over (a, ∞)

Enter f(x), a lower limit a, and the number of nodes to evaluate the half-infinite integral over (a, ∞) accurately with the double exponential (DE) formula.

Input

Enter the integrand f(x), the lower limit a, and the number of nodes to evaluate the definite integral over the half-infinite interval (a, ∞) with the double exponential (DE) formula. f(x) is assumed to decay fast enough as x→∞.

Integrand f(x)

e.g. exp(-x), 1/(x^2), 2x exp(-x^2). Variable x, constants pi/e, functions sin/cos/exp/log/sqrt, etc. Implicit multiplication (2x) is allowed.

Lower limit a

Lower end of the interval (a, ∞).

One-sided node count n

Takes ±n nodes of t centered at 0, using 2n+1 points in total (5 to 2000).

Result

Integral ∫_a^∞ f(x) dx

Lower limit a

Node count

401

Step size h

0.02

Computation details

Lower limit a	0
Upper limit	∞
Node count (2n+1)	401
Step size h	0.02
Skipped nodes	0

How it works

This tool evaluates the definite integral ∫_a^∞ f(x)dx over the half-infinite interval (a, ∞) using the double exponential (DE) formula.
With the change of variables x = a + exp((π/2)·sinh t), the integration range maps to the whole real line (−∞, ∞). The Jacobian is dx/dt = exp((π/2)·sinh t)·(π/2)·cosh t.
The transformed integrand g(t)=f(x(t))·(dx/dt) is approximated by the equally spaced trapezoidal rule with step size h. Because the integrand decays double exponentially toward both ends, high accuracy is obtained even with a finite truncation width.
A larger node count means a finer step size h and higher accuracy, at the cost of more computation. Increase the node count until the result stabilizes.
The integrand f(x) is assumed to decay fast enough as x→∞ so that the integral converges. Slowly decaying or persistently oscillating functions can produce large errors.
Expressions are parsed by a built-in recursive-descent parser. Supported operations are + − × ÷ ^, parentheses, unary minus, and implicit multiplication (e.g. 2x). The variable is x, constants are pi and e, and functions include sin cos tan asin acos atan sinh cosh tanh exp log(=ln) ln log10 sqrt cbrt abs.
The result is a numerical approximation that contains round-off and truncation error. For critical use, confirm convergence across several node counts.

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