How do you use the integral test to determine whether #int dx/(x+lnx)# converges or diverges from #[2,oo)#?

Question

William Abdalla · Accepted Answer

The integral:

#int_2^oo dx/(x+lnx)#

diverges.

As in the interval #x in [2,+oo)# the function: #f(x) = 1/(x+lnx)# is positive and decreasing, and: #lim_(x->oo) f(x) = 0# #f(n) = 1/(n+lnn)# based on the integral test the convergence of the improper integral: #int_2^oo dx/(x+lnx)# is equivalent to the convergence of the series: #sum_(n=2)^oo 1/(n+lnn)# Consider now the harmonic series: #sum_(n=0)^oo 1/n = oo# that we know to be divergent and apply the limit comparison test: #lim_(n->oo) (1/(n+lnn))/(1/n) = lim_(n->oo) (1/(1+lnn/n)) = 1# as the limit is finite the two series have the same character, then also the series #sum_(n=2)^oo 1/(n+lnn)# is divergent.

William Abdalla · Answer

undefined To use the integral test to determine whether $\int_{2}^{\infty} \frac{dx}{x+\ln(x)}$ converges or diverges, we follow these steps:

1. Verify that the function $\frac{1}{x+\ln(x)}$ is continuous, positive, and decreasing for $x \geq 2$.
2. Integrate the function over the interval $[2, \infty)$.
3. Determine if the resulting integral converges or diverges.

Let's go through each step:

1. The function $\frac{1}{x+\ln(x)}$ is continuous and positive for $x \geq 2$. To verify that it is decreasing, we can take its derivative:

$$\frac{d}{dx} \left(\frac{1}{x+\ln(x)}\right) = \frac{-1 - \frac{1}{x}}{(x+\ln(x))^2}$$

The derivative is negative for $x \geq 2$, indicating that the function is decreasing.

2. Integrate the function over the interval $[2, \infty)$:

$$\int_{2}^{\infty} \frac{dx}{x+\ln(x)}$$

3. Now, to determine convergence, we'll analyze the integral:

$$\int_{2}^{\infty} \frac{dx}{x+\ln(x)}$$

We can't find an antiderivative for this function, so we'll compare it to a known function whose integral we can evaluate. Notice that $\frac{1}{x+\ln(x)} < \frac{1}{x}$ for $x \geq 2$.

Therefore, we can compare the integral to the integral of $\frac{1}{x}$ over the same interval:

$$\int_{2}^{\infty} \frac{dx}{x}$$

This integral is a known improper integral, which diverges (as it is the integral of a p-series with $p = 1$).

Since $\frac{1}{x+\ln(x)} < \frac{1}{x}$ and the integral of $\frac{1}{x}$ diverges, by the comparison test, the integral $\int_{2}^{\infty} \frac{dx}{x+\ln(x)}$ also diverges.

Therefore, the integral $\int_{2}^{\infty} \frac{dx}{x+\ln(x)}$ diverges.