## Section 2.10 | The Integral Comparison Theorem

While it may seem to you that evaluating an improper integral (or determining that it is divergent) is near equivalent to evaluating integrals using our previous integration techniques, improper integrals often afford us a greater degree of flexibility. Why?

It is often the case that we need to know whether or not an improper integral (or even a proper integral) converges and we do not necessarily need to know the actual value of the integral. The following theorem states that, by comparing the integral of an original function, say \(f(x)\), to the integral of a somewhat related function, say \(g(x)\), we may be able to determine whether the original function converges or diverges.

It is often the case that we need to know whether or not an improper integral (or even a proper integral) converges and we do not necessarily need to know the actual value of the integral. The following theorem states that, by comparing the integral of an original function, say \(f(x)\), to the integral of a somewhat related function, say \(g(x)\), we may be able to determine whether the original function converges or diverges.

**THEOREM | INTEGRAL COMPARISON THEOREM**

Suppose that \(f\) and \(g\) are continuous functions and that \(f(x) \geq g(x) \geq 0\) on \([a,\infty)\).

- If \(\int_a^{\infty}{f(x) dx}\) is convergent, then \(\int_a^{\infty}{g(x) dx}\) is convergent.
- If \(\int_a^{\infty}{g(x) dx}\) is divergent, then \(\int_a^{\infty}{f(x) dx}\) is divergent.

That is, if \(f(x)\) looks terribly difficult to integrate, but we can find a function \(g(x)\) that is much simpler to integrate, and if we know some ordering relationship between \(f\) and \(g\), we might just be able to determine if the integral of that ugly-looking function \(f\) converges or diverges.

**EXAMPLE 6 | Abounding we will go**

Determine if the integral converges or diverges. \[\int_1^{\infty}{\frac{1}{x^3 + 7x^2 + 2x + 1} dx}\]

**EXAMPLE 7 | Exponential bind**

Determine if the integral converges or diverges. \[\int_3^{\infty}{\frac{1}{x + e^x} dx}\]

**EXAMPLE 8 | A whole lot of noise**

Determine if the integral converges or diverges. \[\int_1^{\infty}{\frac{1 + \sin^4{(2x^2 - \pi)}}{\sqrt{x}} dx}\]

**EXAMPLE 9 | See? Can't we call it quits?**

Determine if the integral converges or diverges. \[\int_0^1{13\frac{\sec^2{(x)}}{x\sqrt{x}} dx}\]

**EXAMPLE 10 | Capturing infinity**

The curve \(y = \frac{1}{x}\) to the right of \(x = 1\) encloses an infinite area; however, if we rotate this about the

*x*-axis, the volume of the resulting, infinitely long solid is finite. How so?