Question

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.\(\int_{3}^{5} \frac{5}{(x-4)^{2}} d x\)

Short answer

The integral \( \int_{3}^{5} \frac{5}{(x-4)^{2}} \, dx \) diverges.

Step-by-step solution

Analyze the Type of Integral

The integral we need to solve is \( \int_{3}^{5} \frac{5}{(x-4)^{2}} \, dx \). This is an improper integral because the integrand \( \frac{5}{(x-4)^{2}} \) becomes undefined at \( x = 4 \), which is within the interval of integration \([3, 5]\).

Split the Integral

Since the point of discontinuity at \( x=4 \) is within the interval, we need to split the integral at this point and evaluate each part separately. We have:\[ \int_{3}^{5} \frac{5}{(x-4)^{2}} \, dx = \int_{3}^{4} \frac{5}{(x-4)^{2}} \, dx + \int_{4}^{5} \frac{5}{(x-4)^{2}} \, dx \]

Evaluate the Improper Integral from 3 to 4

Now we evaluate the first part of the integral which is improper due to the discontinuity at 4:\[ \lim_{b \to 4^{-}} \int_{3}^{b} \frac{5}{(x-4)^{2}} \, dx \]To solve, use the antiderivative \( -\frac{5}{x-4} \). Then,\[ = \lim_{b \to 4^{-}} \left[-\frac{5}{x-4}\right]_{3}^{b} = \lim_{b \to 4^{-}} \left(-\frac{5}{b-4} + \frac{5}{3-4}\right) = \lim_{b \to 4^{-}} \left(-\frac{5}{b-4} - 5\right) \]As \( b \to 4^{-} \), \( -\frac{5}{b-4} \to -\infty \), which results in divergence to \(-\infty\).

Conclusion of Convergence or Divergence

Since the integral from 3 to 4 diverges, there is no need to evaluate the integral from 4 to 5 separately. When any part of a split improper integral diverges, the whole integral diverges. Thus, the original integral does not converge.

Key concepts

Convergence

In calculus, convergence refers to the behavior of an integral as it approaches a certain limit. When we talk about the convergence of improper integrals, we mean whether the integral evaluates to a finite number. To determine convergence, we often need to handle limits, especially when dealing with integrals that have points of discontinuity or infinities.

• One common scenario is when an integral has a discontinuity within its limits, as seen in the exercise where the denominators go to zero.
• Another occurs when the function being integrated, called the integrand, approaches infinity.

When an integral converges, it means that despite any unexpected behavior within the integration limits, the result "settles" to a finite number. Evaluating convergence usually involves separating or "splitting" the integral at the points where these behaviors occur, also known as points of discontinuity. If all parts of this split integral converge individually, the initial integral is deemed to converge as well. Otherwise, if any section diverges, the whole integral diverges.

Divergence

Divergence is essentially the opposite of convergence. An integral diverges if it does not settle at a finite value and instead approaches an infinite value or simply doesn't resolve at all. This typically happens when:

• The limits of integration include a point where the integrand becomes infinite.
• There is a discontinuity within the integration limits, as evidenced in the given example where the integrand is undefined at the point \( x = 4 \).

In the exercise provided, the integral diverges because when approaching \( x = 4 \) from either side, the function heads towards infinity \((-\frac{5}{x-4})\). This divergence is crucial: it signifies that the integral does not yield a well-defined area under the curve between the specified limits. Thus, even if part of a split integral appears to converge, such as when separately evaluated, having just one divergent part implies the integral as a whole cannot be resolved to a finite number.

Antiderivatives

Antiderivatives are central to solving integrals, both proper and improper. An antiderivative of a function is another function whose derivative gives the original function. It allows us to evaluate definite integrals over an interval, by using the Fundamental Theorem of Calculus.

• Finding an antiderivative can transform a complex integral into a simpler calculation, by taking the difference of the antiderivative's values at the bounds of the integral.
• In our example, the antiderivative of \( \frac{5}{(x-4)^2} \) is determined as \(-\frac{5}{x-4}\).

Using antiderivatives involves evaluating the limit, especially when addressing integrals with undefined points or infinite limits, transforming them from a regular mathematical expression into a form suitable for solving.
Understanding antiderivatives extends beyond just computing them; it's about grasping how they allow improper integrals like those in the example to be approached methodically and systematically. By integrating these parts across different bounds and employing limits, we can assess whether results converge or diverge effectively.