determine whether the integral is convergent or divergent. if it is convergent, evaluate it. (if the…

determine whether the integral is convergent or divergent. if it is convergent, evaluate it. (if the quantity diverges, enter diverges.)\n int_{1}^{infty}\frac{37}{ln(x)}dx \n\nresources\nread it\nwatch it\n\nenhanced feedback\nplease try again. first evaluate the integral from 1 to t. determine whether or not this is convergent as t approaches (infty). if it is, evaluate the limit of the integral.

determine whether the integral is convergent or divergent. if it is convergent, evaluate it. (if the quantity diverges, enter diverges.)\n int_{1}^{infty}\frac{37}{ln(x)}dx \n\nresources\nread it\nwatch it\n\nenhanced feedback\nplease try again. first evaluate the integral from 1 to t. determine whether or not this is convergent as t approaches (infty). if it is, evaluate the limit of the integral.

Answer

Explanation:

Step1: Use substitution

Let $u = \ln(x)$, then $du=\frac{1}{x}dx$. When $x = 1$, $u=\ln(1) = 0$; when $x=t$, $u=\ln(t)$. The integral $\int_{1}^{t}\frac{37}{\ln(x)}\cdot\frac{1}{x}dx$ becomes $\int_{0}^{\ln(t)}37du$.

Step2: Evaluate the integral

$\int_{0}^{\ln(t)}37du=37u\big|_{0}^{\ln(t)}=37\ln(t)-37\times0 = 37\ln(t)$.

Step3: Find the limit as $t\to\infty$

We want to find $\lim_{t\to\infty}\int_{1}^{t}\frac{37}{x\ln(x)}dx=\lim_{t\to\infty}37\ln(t)$. As $t\to\infty$, $\ln(t)\to\infty$, so $37\ln(t)\to\infty$.

Answer:

DIVERGES