Question

Rewrite \(\int_{2}^{\infty} \frac{d x}{x^{1 / 5}}\) as a limit and then show that the integral diverges.

Short answer

Based on the provided step by step solution, the integration \(\int_{2}^{\infty}\frac{dx}{x^{1/5}}\) diverges.

Step-by-step solution

Rewrite the integral as a limit

We can rewrite the given improper integral as a limit: \(\int_{2}^{\infty}\frac{dx}{x^{1/5}} = \lim_{a \to \infty}\int_{2}^{a}\frac{dx}{x^{1/5}}\)

Find the antiderivative of the integrand

Now we need to find the antiderivative of the integrand, which is \(\frac{1}{x^{1/5}}\). To do this, we can use the power rule for antiderivatives, which states that \(\int x^n dx = \frac{x^{n+1}}{n+1}\), where \(n\) is any real number: \(\int\frac{dx}{x^{1/5}} = \int x^{-1/5} dx= \frac{x^{-1/5+1}}{-1/5+1}=5x^{4/5} + C\)

Apply the fundamental theorem of calculus (FTC)

Now, we will apply the fundamental theorem of calculus by evaluating the definite integral with the established antiderivative, and taking the limit as \(a\) approaches infinity: \(\lim_{a \to \infty}\int_{2}^{a}\frac{dx}{x^{1/5}} = \lim_{a \to \infty} \left(5a^{4/5} - 5\cdot2^{4/5}\right)\)

Evaluate the limit

Lastly, let's evaluate the limit: \(\lim_{a \to \infty} \left(5a^{4/5} - 5\cdot2^{4/5}\right) = \infty - 5\cdot2^{4/5} = \infty\) Since the limit is infinite, the integral diverges.

Key concepts

Antiderivative Calculation

Understanding antiderivative calculation is essential when dealing with integrals, whether they are proper or improper. An antiderivative of a function, sometimes called an indefinite integral, is essentially the reverse process of differentiation. If you have a rate of change, the antiderivative provides the original function before that change was applied.

Let's take the function \( \frac{1}{x^{1/5}} \) as an example. An antiderivative of this function will give us a function that, when differentiated, returns \( \frac{1}{x^{1/5}} \). The calculated antiderivative is crucial because it allows us to evaluate definite integrals over an interval, which in turn can tell us things like the accumulated change in a certain context, such as area under a curve.

Power Rule for Antiderivatives

The power rule for antiderivatives is a simple and powerful tool in calculus.

For any real number \(n \eq -1\), the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), plus a constant of integration \(C\).

When dealing with \(\frac{dx}{x^{1/5}}\), the function can be rewritten as \(x^{-1/5}\) and applying the power rule gives us the antiderivative \(5x^{4/5} + C\). This step is fundamental for evaluating improper integrals; the constant \(C\) falls away when calculating definite integrals, but for indefinite integrals, it represents the family of all possible antiderivatives.

Fundamental Theorem of Calculus

The fundamental theorem of calculus is a bridge between differentiation and integration. It comes in two parts: the first connects the concept of the antiderivative with the definite integral, and the second allows for the evaluation of a definite integral by using an antiderivative.

In the context of our problem, we apply the second part. Once we have the antiderivative of our function, we can use it to evaluate the definite integral from 2 to \(a\). Mathematically, this translates to the definite integral being equal to the difference of the antiderivative evaluated at the upper and lower limits of our integral. Put simply, if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) dx = F(b) - F(a)\).

Limit Evaluation

When we deal with improper integrals like \(\int_{2}^{\infty} \frac{dx}{x^{1/5}}\), we need to use limit evaluation to understand the behavior of the function as it approaches a boundary that is not finite—in this case, infinity.

In following our steps, after using the antiderivative method, we replaced the infinite boundary with a variable \(a\), and took the limit as \(a\) goes to \( \infty \). This step is pivotal to determining whether the area under the curve is finite or not. If the limit evaluates to an infinite number, as it does in our example (\(\lim_{a \to \infty} (5a^{4/5}) = \infty \)), then the original integral diverges. This means that the area under the curve, from the lower limit to infinity, is infinite.