determine whether the following integral converges or diverges. ∫0,∞ dx/(e^x + 2x) as x approaches ∞, which…

Explanation:

Step1: Analyze the integrand as $x\to\infty$

As $x\to\infty$, the exponential - function $e^{x}$ dominates the linear function $2x$. So, $e^{x}+2x\approx e^{x}$ for large $x$. We know that for all $x\geq0$, $e^{x}+2x > e^{x}$. Then, $\frac{1}{e^{x}+2x}<\frac{1}{e^{x}}$.

Step2: Recall the Comparison Test for improper integrals

The Comparison Test for improper integrals states that if $0\leq f(x)\leq g(x)$ for all $x\geq a$ and $\int_{a}^{\infty}g(x)dx$ converges, then $\int_{a}^{\infty}f(x)dx$ also converges. Consider the integral $\int_{0}^{\infty}\frac{1}{e^{x}}dx$. We know that $\int_{0}^{\infty}e^{-x}dx=\lim_{b\to\infty}\int_{0}^{b}e^{-x}dx$. Using the antiderivative of $e^{-x}$ which is $-e^{-x}$, we have $\lim_{b\to\infty}(-e^{-x})\big|{0}^{b}=\lim{b\to\infty}(-e^{-b}+e^{0}) = 1$. Since $\int_{0}^{\infty}\frac{1}{e^{x}}dx$ converges and $\frac{1}{e^{x}+2x}<\frac{1}{e^{x}}$ for $x\geq0$, by the Comparison Test, $\int_{0}^{\infty}\frac{1}{e^{x}+2x}dx$ converges.

For the first multiple - choice question: Since $e^{x}+2x > e^{x}$ for $x > 0$, then $\frac{1}{e^{x}+2x}<\frac{1}{e^{x}}$. The correct option is D.

For the second multiple - choice question: Since $\frac{1}{e^{x}+2x}<\frac{1}{e^{x}}$ and $\int_{0}^{\infty}\frac{1}{e^{x}}dx$ converges, by the Comparison Test for improper integrals, $\int_{0}^{\infty}\frac{dx}{e^{x}+2x}$ converges. The correct option is B.

Answer:

For the first multiple - choice question: D. $\frac{1}{e^{x}+2x}<\frac{1}{e^{x}}$ For the second multiple - choice question: B. $\int_{0}^{\infty}\frac{dx}{e^{x}+2x}$ converges.