Chapter 8.8, Problem 52E

Comparison Test for Improper Integrals In somecases, it is impossible to find the exact value of an improperintegral, but it is important to determine whether the integralconverges or diverges. Suppose the functions f and g arecontinuous and $0\le g(x)\le f(x)$ on the interval $[a,\infty )$. It canbe shown that if ${\displaystyle {\int }_a^{\infty }f(x)dx}$ converges, then ${\displaystyle {\int }_a^{\infty }g(x)dx}$ alsoconverges, and if ${\displaystyle {\int }_a^{\infty }f(x)dx}$ diverges, then ${\displaystyle {\int }_a^{\infty }g(x)dx}$ alsodiverges. This is known as the Comparison Test for improperintegrals.

(a) Use the Comparison Test to determine whether ${\displaystyle {\int }_a^{\infty }g(x)dx}$

converges or diverges. (Hint: Use the fact that $e^{-x^2}\le e^{-x}$

for $x\ge 1$

.)

(b) Use the Comparison Test to determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{x^5+1}dx}$ converges or diverges. (Hint: Use the fact

that $\frac{1}{x^5+1}\le \frac{1}{x^5}\text{for}x\ge 1$

.)