Improper Integral: Definition & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

When integrating a function, infinity might appear in the limits of integration. The function itself could also go to infinity. These types of integrals are called improper integrals, which we will explore in this lesson.

What Are Improper Integrals?

Infinity in math is when something keeps getting bigger without limit. This can happen in the lower or upper limits of an integral, or both. An integral is the area under a curve, and integrating a function gives us the area under the curve of the function by summing small portions of area. An integral is represented as S or the symbol:

Symbol representing an integral
Symbol representing an integral

Infinity can also happen if the function being integrated, the integrand, gets very large for values between the lower and upper limits of integration. Integrals that involve infinity are called improper integrals. These types of integrals cannot be solved in the usual way.

Integrating at Infinity

Let's say we want to know the area under a curve from some fixed value of x all the way to infinity. We can integrate out to some variable b and then let b go to infinity. Our first example uses 1/x squared as the function for the curve.

To calculate the area under this curve from x = 1 to x = b we write:

integral of 1/x^2 from 1 to b

This area A is shown shaded below.

Area under the 1/x^2 curve from 1 to b
graph of area under the 1/x^2 curve from 1 to b

Moving b further and further to the right, we collect more and more area. Eventually, we will have all the area from 1 to infinity. Before considering how to deal with x at infinity, let's write A as a function of b.

To do this, use:

x^2 is 1/x^(-2)

Then, the anti-derivative of x^(-2) is:

the anti-derivative of x^(-2)

Our integral becomes:

integral result is 1 - 1/b

Now, we have the area in terms of how far to the right we integrate. If b is equal to 2, the area A is 1 - 1/b = 1 - ½ = ½. To integrate up to infinity, we need much bigger numbers for b. What if b = 100? Then A = 1 - 1/100 = .990.

For b = 1000, A = 1 - 1/1000 = .999. We say the area is approaching the value of 1 as b goes to infinity. Since the answer is a number, we say the integral converges to that number. In this example, the integral converges to 1.

Here is a more formal expression that uses the limit concept:

b going to infinity replaced by the limit

This is exactly what we did. We replaced the infinity with a variable b. Then, after obtaining an expression for the integral with this variable b, we let b go to infinity.

You might think improper integrals always converge if the function goes to zero at infinity. But they don't really!

Area under the curve of 1/x^(1/2) from 1 to b.
graph of area under the curve of 1/x^(1/2)

The function here is one over the square root of x. First, write:

1/sqrt(x) is x^(-1/2)

Then, the anti-derivative of x^(-1/2) is:

anti-derivative of x^(-1/2) is 2x^(1/2)

The area A from x = 1 to x = b is given by:

the area is 2sqrt(b) - 2

As b gets larger, the area A keeps getting larger. In fact, for b equal to infinity, A must be infinity. Thus, the area approaches infinity as x goes to infinity. The area under this curve is not a number. The area is infinite. The integral does not converge. When the limit of an integral is infinite or does not exist, then we say the integral diverges.

Now let's plot our functions on the same graph:

Plots of x^(-1/2) (pink) and x^(-2) (blue)
plots of x^(-1/2) and x^(-2)

As you can see, both functions go to zero at infinity but only one of the integrals converges. Why is that? The x^(-1/2) function does not go to zero fast enough.

Dealing With Integrands Going to Infinity

Sometimes, the integrand will go to infinity inside the interval of integration. Remember, the integrand is the function we are integrating. Here is an example integrating from x = 0 to x = 1. There is an issue at x = 0 where the integrand goes to infinity.

Area under the curve from a to 1.
area under the curve from a to 1

The limit in this case is called a right-sided limit. We are approaching the value of 0 from the right side of 0. To indicate this we write 0+ . This means the value of a will always be positive. We want to avoid taking the square root of a negative number.

To find the area A we write:

using the right-sided limit

At a = 0, the square root of 0 is zero. This gives us an area of A = 2.

Applying Integers to a Real-Life Example

In the North, it's common to burn wood in a fireplace for heat. We add wood to the stockpile of wood each day, and the number of logs left depends on how cold it is outside. As the weather gets colder, we burn more wood. The stockpile gets smaller, approaching zero as the season goes on. And winter is long! Fortunately, there are limits to the winter season.

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