evaluate the following integral or state that it diverges. ∫8,∞ 4 dy / (y ln y)

Explanation:

Step1: Use substitution

Let $u = \ln y$, then $du=\frac{1}{y}dy$. When $y = 8$, $u=\ln 8$; as $y\rightarrow\infty$, $u\rightarrow\infty$. The integral $\int_{8}^{\infty}\frac{4dy}{y\ln y}$ becomes $4\int_{\ln 8}^{\infty}\frac{du}{u}$.

Step2: Evaluate the integral

The antiderivative of $\frac{1}{u}$ is $\ln|u|$. So, $4\int_{\ln 8}^{\infty}\frac{du}{u}=4\lim_{b\rightarrow\infty}\int_{\ln 8}^{b}\frac{du}{u}=4\lim_{b\rightarrow\infty}(\ln u|_{\ln 8}^{b})$.

Step3: Calculate the limit

$4\lim_{b\rightarrow\infty}(\ln u|{\ln 8}^{b})=4\lim{b\rightarrow\infty}(\ln b-\ln(\ln 8))$. Since $\lim_{b\rightarrow\infty}\ln b=\infty$, the value of the limit is $\infty$.

Answer:

The integral diverges.