Question

92. Integral with a parameter For what values of p does the integral∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?

Accepted Answer

First, rewrite the integral to clearly identify the integrand and the limits: consider the integral $$ \int_1^{\infty} \frac{1}{x (\ln x)^p} \, dx . To $$analyze convergence, perform the substitution $$ t = \ln x . $$Then, $$ dt = \frac{1}{x} dx $$, which implies $$ dx = x \, dt . $$Substitute into the integral: $$ \int_1^{\infty} \frac{1}{x (\ln x)^p} \, dx = \int_0^{\infty} \frac{1}{t^p} \, dt $$ Now, examine the integral $$ \int_0^{\infty} t^{-p} \, dt $$ for convergence. Since the integral is improper at both limits, analyze the behavior near 0 and near infinity separately: - Near $$ t = 0 $$, the integral $$ \int_0^1 t^{-p} \, dt $$ converges if and only if $$ p  1 . $$Since both conditions cannot be true simultaneously, the integral converges only if $$ p \u003e 1 $$ because the original integral's lower limit corresponds to $$ t = 0 $$ and the upper limit to $$ t = \infty . $$Actually, the substitution shows the lower limit is $$ t=0 $$, so we must be careful: the integral converges if $$ p \u003e 1 $$ because the integral near infinity converges only then, and near zero the integral converges for $$ p  1 $$, compute the value of the integral by evaluating $$ \int_0^{\infty} t^{-p} \, dt $$ with proper limits and express the result in terms of $$ p .$$

92. Integral with a parameter For what values of p does the integral
∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?

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Key Concepts

Improper Integrals and Convergence

Behavior of the Integrand Near Infinity

Use of Substitution in Evaluating Integrals