Problem 32 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \arctan x d x\)

Step-by-Step Solution

Answer

The integral \(\int_{1}^{\infty} \arctan x d x\) is divergent.

Step 1: Applying the limit

To start, take the limit of \(\arctan x\) as \(x\) approaches infinity, i.e., \(\lim _{x \rightarrow \infty} \arctan x\).

Step 2: Interpretation of arctangent

In mathematics, the arc tangent of infinity, \(\arctan(\infty)\), is equal to \(\pi/2\). Therefore, computing the original limit results in \(\pi/2\), which does not equal to zero.

Step 3: Determine convergence

Then, by the given condition, because the limit as \(x\) approaches infinity of \(\arctan x\) is not 0, the integral \(\int_{1}^{\infty} \arctan x d x\) is divergent.

Key Concepts

Convergence and Divergence

When studying improper integrals, an essential concept to grasp is whether these integrals converge or diverge. **Convergence** of an integral occurs when the area under the curve of a function is finite, meaning it adds up to a specific value. **Divergence**, however, implies that the area becomes infinite or does not settle to any specific value.
For an improper integral like \( \int_{a}^{\infty} f(x) \, dx \), an effective way to decide on convergence or divergence is by evaluating the limit of the function as \( x \) approaches infinity. If \( \lim_{x \rightarrow \infty} f(x) = 0 \), then the function potentially converges provided the series itself does not blow up. If the limit is not zero, like when it equals a constant such as \( \pi/2 \), the integral is definitely divergent.
This decision helps in determining whether further evaluation of the integral is productive.

Arctangent Function

The arctangent function, denoted as \( \arctan x \), is a vital mathematical concept, especially when dealing with trigonometric integration and improper integrals. This function essentially reveals the angle whose tangent is \( x \). It is the inverse operation of the tangent function and is often used in calculus to find angles when given certain ratios.
Unlike other functions, the range of \( \arctan x \) is limited between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). Therefore, regardless of how large \( x \) becomes, \( \arctan x \) will never exceed this range.
For this reason, if you compute \( \lim_{x \rightarrow \infty} \arctan x \), you arrive at \( \frac{\pi}{2} \). In the context of improper integrals, if the outcome of such a limit is not zero, it clearly indicates that the integral is divergent.

Limit at Infinity

The notion of the limit at infinity plays a crucial part in determining whether an improper integral converges or diverges. In calculus, taking the limit as \( x \) approaches infinity helps in understanding the behavior of functions for extremely large values.
Specifically, when evaluating \( \lim_{x \rightarrow \infty} f(x) \), you're essentially observing the end behavior of \( f(x) \). Whether it approaches a constant, zero, or grows continually can significantly affect the convergence of an integral.

• If the limit equals zero, the function is a candidate for convergence, although additional checks on the nature of the integral are required.
• If the limit approaches a non-zero constant or diverges itself, like going to infinity, the integral will not converge.

In practical exercises, like the one stated, calculating limits at infinity helps confirm the conclusions regarding convergence and can guide whether to continue evaluating the integral.