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Improper Integral


An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.

For example, the integral

 int_1^inftyx^(-2)dx
(1)

is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values

 int_1^yx^(-2)dx=1-1/y
(2)

and then taking the limit as y->infty,

int_1^inftyx^(-2)dx=lim_(y->infty)int_1^yx^(-2)dx
(3)
=lim_(y->infty)1-1/y
(4)
=1.
(5)

Improper integrals of the form

 int_a^bf(x)dx
(6)

with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If f(x) decreases at least as fast as 1/x^2, then let

t=1/x
(7)
dt=-(dx)/(x^2)
(8)
dx=-x^2dt
(9)
=-(dt)/(t^2),
(10)

and

int_a^bf(x)dx=-int_(1/a)^(1/b)1/(t^2)f(1/t)dt
(11)
=int_(1/b)^(1/a)1/(t^2)f(1/t)dt.
(12)

If f(x) diverges as (x-a)^gamma for gamma in [0,1], let

x=t^(1/(1-gamma))+a
(13)
dx=1/(1-gamma)t^((1/1-gamma)-1)dt
(14)
=1/(1-gamma)t^([1-(1-gamma)]/(1-gamma))dt
(15)
=1/(gamma-1)t^(gamma/(1-gamma))dt
(16)
t=(x-a)^(1-gamma),
(17)

and

 int_a^bf(x)dx=1/(1-gamma)=int_0^((b-a)^(1-gamma))t^(gamma/(1-gamma))f(t^(1/(1-gamma))+a)dt.
(18)

If f(x) diverges as (x+b)^gamma for gamma in [0,1], let

x=b-t^(1/(1-gamma))
(19)
dx=-1/(gamma-1)t^(gamma/(1-gamma))dt
(20)
t=(b-x)^(1-gamma),
(21)

and

int_a^bf(x)dx=1/(1-gamma)
(22)
=int_0^((b-a)^(1-gamma))t^(gamma/(1-gamma))f(b-t^(1/(1-gamma)))dt.
(23)

If the integral diverges exponentially, then let

t=e^(-x)
(24)
dt=-e^(-x)dx
(25)
x=-lnt,
(26)

and

 int_a^inftyf(x)dx=int_0^(e^(-a))f(-lnt)(dt)/t.
(27)

See also

Definite Integral, Integral, Proper Integral, Singular Integral

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References

Jeffreys, H. and Jeffreys, B. S. "Infinite and Improper Integrals." §1.104 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 33-34, 1988.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Improper Integrals." §4.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 135-140, 1992.

Referenced on Wolfram|Alpha

Improper Integral

Cite this as:

Weisstein, Eric W. "Improper Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ImproperIntegral.html

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