**Objective**: Solve the different types of improper integrals

**Definition.**An improper integral is an integral where one or the two limits of integration are infinite. An integral is also improper when the function has an infinite discontinuity. Let's look at the two scenarios:

**I. One or two of the limits of integration are infinite.**

**a. One of the limit of integration is finite**,

If f is continuous over the interval [a, +∞[, then

If the limit is finite the integral converges. If the limit is infinite the integral diverges.

**Geometric Interpretation**

The integral represents the area between the verticals passing by a and l. As l approaches infinity, the area under the curve spreads to infinity. It's fair to say that this situation represents .

We can define as the limit of when l approaches infinity.

We can then set a rule to calculate the improper integral where the lower limit is finite and the upper limit is infinite. In this case we have to replace the infinite limit by l and find the limit of when l approaches infinity.

**Example 1.**Calculate

**Solution**

**Let's replace infinity by l and calculate the limit of the finite integral when l approaches infinity.**

We have:

**b. The two limits of integration are infinite**

In this case we choose any value a in the interval ]-∞, +∞[ and calculate the integral over the sub-intervals

]-∞, a[∪ ]a , +∞[

**Solution**

Let's split the integral over the intervasl ]-,0] and [0,+[:

Let's evaluate the first integral on the right side:

Let's calculate the limit by substituting l by -∞:

Let's evaluate the second integral:

**II The function to integrate is discontinued.**

Let's say that we have to calculate an integral over an interval ]a, b[ where the function f is continued. If the function isn't continued at a point c of the interval, we evaluate the integral over the intervals ]a,c[∪]c,d[.

Example. Evaluate

**Solution**

**The function is discontinued at x = 1. Therefore we integrate it over the 2 intervals ]0,1[∪ ]1.4[.**

Let's calculate the first integral:

= ln │1⁻-1│-ln1

= ln o⁺

= -∞

**Exercises**

Evaluate. Tell if the integral converges or diverges:

If you want to learn more about Calculus check this site Center for Integral Development

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