# integrate

##### Integration of One-Dimensional Functions

Adaptive quadrature of functions of one variable over a finite or infinite interval.

##### Usage

```
integrate(f, lower, upper, …, subdivisions = 100L,
rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)
```

##### Arguments

- f
an R function taking a numeric first argument and returning a numeric vector of the same length. Returning a non-finite element will generate an error.

- lower, upper
the limits of integration. Can be infinite.

- …
additional arguments to be passed to

`f`

.- subdivisions
the maximum number of subintervals.

- rel.tol
relative accuracy requested.

- abs.tol
absolute accuracy requested.

- stop.on.error
logical. If true (the default) an error stops the function. If false some errors will give a result with a warning in the

`message`

component.- keep.xy
unused. For compatibility with S.

- aux
unused. For compatibility with S.

##### Details

Note that arguments after `…`

must be matched exactly.

If one or both limits are infinite, the infinite range is mapped onto a finite interval.

For a finite interval, globally adaptive interval subdivision is used in connection with extrapolation by Wynn's Epsilon algorithm, with the basic step being Gauss--Kronrod quadrature.

`rel.tol`

cannot be less than ```
max(50*.Machine$double.eps,
0.5e-28)
```

if `abs.tol <= 0`

.

In R versions \(\le\) 3.2.x, the first entries of
`lower`

and `upper`

were used whereas an error is signalled
now if they are not of length one.

##### Value

A list of class `"integrate"`

with components

the final estimate of the integral.

estimate of the modulus of the absolute error.

the number of subintervals produced in the subdivision process.

`"OK"`

or a character string giving the error message.

the matched call.

##### Note

Like all numerical integration routines, these evaluate the function on a finite set of points. If the function is approximately constant (in particular, zero) over nearly all its range it is possible that the result and error estimate may be seriously wrong.

When integrating over infinite intervals do so explicitly, rather than just using a large number as the endpoint. This increases the chance of a correct answer -- any function whose integral over an infinite interval is finite must be near zero for most of that interval.

For values at a finite set of points to be a fair reflection of the behaviour of the function elsewhere, the function needs to be well-behaved, for example differentiable except perhaps for a small number of jumps or integrable singularities.

`f`

must accept a vector of inputs and produce a vector of function
evaluations at those points. The `Vectorize`

function
may be helpful to convert `f`

to this form.

##### References

R. Piessens, E. deDoncker--Kapenga, C. Uberhuber, D. Kahaner (1983)
*Quadpack: a Subroutine Package for Automatic Integration*;
Springer Verlag.

##### Examples

`library(stats)`

```
# NOT RUN {
integrate(dnorm, -1.96, 1.96)
integrate(dnorm, -Inf, Inf)
## a slowly-convergent integral
integrand <- function(x) {1/((x+1)*sqrt(x))}
integrate(integrand, lower = 0, upper = Inf)
## don't do this if you really want the integral from 0 to Inf
integrate(integrand, lower = 0, upper = 10)
integrate(integrand, lower = 0, upper = 100000)
integrate(integrand, lower = 0, upper = 1000000, stop.on.error = FALSE)
## some functions do not handle vector input properly
f <- function(x) 2.0
try(integrate(f, 0, 1))
integrate(Vectorize(f), 0, 1) ## correct
integrate(function(x) rep(2.0, length(x)), 0, 1) ## correct
## integrate can fail if misused
integrate(dnorm, 0, 2)
integrate(dnorm, 0, 20)
integrate(dnorm, 0, 200)
integrate(dnorm, 0, 2000)
integrate(dnorm, 0, 20000) ## fails on many systems
integrate(dnorm, 0, Inf) ## works
# }
# NOT RUN {
integrate(dnorm, 0:1, 20) #-> error!
## "silently" gave integrate(dnorm, 0, 20) in earlier versions of R
# }
```

*Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2*