Math Problem Statement
Solution
This problem focuses on evaluating the p-integral and examining its behavior for different values of . The integral in question is:
\begin{cases} \frac{1}{p-1} & \text{if } p > 1, \\ \infty & \text{if } p \leq 1. \end{cases}$$ Let's work through each part: ### Part (a): $$p > 1$$ The integral to evaluate is: $$\int_1^\infty \frac{1}{x^p} \, dx$$ 1. **Find the antiderivative**: $$\int \frac{1}{x^p} \, dx = \int x^{-p} \, dx = \frac{x^{1-p}}{1-p} \quad \text{(valid for } p \neq 1 \text{)}.$$ 2. **Apply limits**: $$\int_1^\infty \frac{1}{x^p} \, dx = \lim_{b \to \infty} \left[ \frac{x^{1-p}}{1-p} \right]_1^b = \lim_{b \to \infty} \left( \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p} \right).$$ - As $$b \to \infty$$, $$b^{1-p} \to 0$$ if $$p > 1$$. - So, the integral becomes: $$\frac{0}{1-p} - \frac{1}{1-p} = \frac{1}{p-1}.$$ - Therefore, the integral **converges** to $$\frac{1}{p-1}$$ when $$p > 1$$. ### Part (b): $$p < 1$$ We now evaluate the same integral, but with $$p < 1$$: $$\int_1^\infty \frac{1}{x^p} \, dx = \lim_{b \to \infty} \left( \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p} \right).$$ - As $$b \to \infty$$, $$b^{1-p} \to \infty$$ because $$1 - p > 0$$ for $$p < 1$$. - This causes the integral to diverge, as $$\lim_{b \to \infty} b^{1-p} = \infty$$. Thus, the integral **diverges** for $$p < 1$$. ### Part (c): $$p = 1$$ For this case, the integral becomes: $$\int_1^\infty \frac{1}{x} \, dx.$$ - The antiderivative of $$\frac{1}{x}$$ is $$\ln x$$, so: $$\int_1^\infty \frac{1}{x} \, dx = \lim_{b \to \infty} \ln b - \ln 1 = \lim_{b \to \infty} \ln b = \infty.$$ Thus, the integral **diverges** for $$p = 1$$. ### Summary: - For $$p > 1$$, the integral converges to $$\frac{1}{p-1}$$. - For $$p \leq 1$$, the integral diverges. Would you like any additional details or have further questions? Here are some related questions you might explore: 1. What happens if we change the lower limit of the integral to something other than 1? 2. How does the convergence of the p-integral relate to improper integrals? 3. Can you prove the result using comparison tests or other methods for improper integrals? 4. What is the physical or geometrical interpretation of the p-integral? 5. How does this integral behave if we integrate over a finite interval, say from $$1$$ to some $$a > 1$$? **Tip**: Always check whether a definite integral is improper, as it helps in evaluating the convergence or divergence.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Improper Integrals
Convergence and Divergence of Integrals
Formulas
∫₁^∞ 1/x^p dx
Antiderivative of x^-p: x^(1-p)/(1-p)
Natural Logarithm for p = 1
Theorems
Fundamental Theorem of Calculus
Improper Integral Test for Convergence
Suitable Grade Level
Undergraduate
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