Evaluation of Improper Integrals

In a Nut Shell: Integrals involving unbounded regions are termed improper integrals.

The integral may be unbounded by having its limits of integration unbounded ( i.e.

the upper limit, the lower limit, or both limits of integration unbounded), by having

the integrand unbounded, or by having both the limits of integration and the integrand

unbounded. Reference these three cases as Type 1, Type 2, and Type 3 as provided

in the tables below.

Strategy: Identify where the integral is unbounded, replace the bound with a

variable, and then evaluate the integral as the variable approaches its limit.

Type 1: Limits of integration are unbounded; can involve ± ∞

∞ t

Example: I = ∫( [1 /x√x]dx I = lim ∫( [1 /x√x]dx

2 t →∞ 2

Here the upper limit is unbounded ( +∞) .

Type 2: Integrand is unbounded

4 4

Example: I = ∫ [1 /x√x]dx I = lim ∫ [1 /x√x]dx

0 t →0 t

Here the integrand is unbounded at x = 0 .

Type 3: Both the limits of integration (in the example below the upper limit)

and the integrand are unbounded (at x = 0).

∞

Example: I = ∫ [1 /(x + x²)]dx

0

In a situation like this one needs to break region of integration into two parts. Call

the integrals I₁ and I₂.

Click here for further discussion of this example.

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