\( \int ^ { \infty } \frac { 1 } { x + 1 } \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{1}^{+\infty} \frac{1}{x+1} dx\) - step1: Rewrite the improper integral: \(\lim _{a\rightarrow +\infty}\left(\int_{1}^{a} \frac{1}{x+1} dx\right)\) - step2: Evaluate the integral: \(\lim _{a\rightarrow +\infty}\left(\ln{\left(\frac{\left|a+1\right|}{2}\right)}\right)\) - step3: Rewrite the expression: \(\ln{\left(\lim _{a\rightarrow +\infty}\left(\frac{\left|a+1\right|}{2}\right)\right)}\) - step4: Calculate: \(\ln{\left(+\infty\right)}\) - step5: Calculate: \(+\infty\) - step6: Calculate: \(\textrm{Diverges}\) The integral \[ \int_{1}^{\infty} \frac{1}{x + 1} \, dx \] is an improper integral. Upon evaluation, it diverges, meaning that it does not converge to a finite value.