Question

77–86. Comparison Test Determine whether the following integrals converge or diverge.78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

Accepted Answer

First, identify the behavior of the integrand $$ \frac{1}{e^{x} + x + 1}  as$$ $$ x 	o \infty $$ and as $$ x 	o 0  to $$understand the nature of the integral $$ \int_0^{\infty} \frac{dx}{e^{x} + x + 1} . $$For large $$ x $$, note that $$ e^{x} $$ grows much faster than $$ x + 1 $$, so the integrand behaves approximately like $$ \frac{1}{e^{x}} . $$This suggests comparing it to the integral $$ \int_0^{\infty} e^{-x} \, dx $$, which is a convergent integral. For $$ x $$ near 0, observe that $$ e^{x} + x + 1  is $$continuous and positive, so the integrand is finite and well-behaved near 0, meaning there is no issue with convergence at the lower limit. Apply the Comparison Test by finding a function $$ g(x) $$ such that $$ 0 \leq \frac{1}{e^{x} + x + 1} \leq g(x) $$ for all $$ x \geq 0 $$, and $$ \int_0^{\infty} g(x) \, dx  is $$known to converge. For example, use $$ g(x) = e^{-x} $$ since $$ e^{x} + x + 1 \u003e e^{x} $$ implies $$ \frac{1}{e^{x} + x + 1} \u003c e^{-x} . $$Since $$ \int_0^{\infty} e^{-x} \, dx $$ converges, by the Comparison Test, the original integral $$ \int_0^{\infty} \frac{dx}{e^{x} + x + 1} $$ also converges.

77–86. Comparison Test Determine whether the following integrals converge or diverge.
78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

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주요 개념

Improper Integrals

Comparison Test for Improper Integrals

Behavior of Exponential Functions at Infinity