Question

1. Use the Comparison Test to determine whether the following integrals converge, diverge or the test is inconclusive. \( (a) \int_2^8 \frac{dx}{x^2 - 4} \) \( (b) \int_0^1 \frac{dx}{x^3 + x^5} \) Hint 1: We have the p-test for improper integrals of the second kind. The proof is exactly like that of the first kind, namely, we apply the limit definition. Here is the statement of the p-test for improper integrals of the second kind. \( \int_0^b \frac{dx}{x^p} = \begin{cases} \text{divergent,} & \text{if } p \ge 1\\ \frac{b^{-p+1}}{-p+1}, & \text{if } p < 1. \end{cases} \) Hint 2: For part (a), use substitution \( u = x - 2 \).

          1. Use the Comparison Test to determine whether the following integrals converge, diverge or
the test is inconclusive.
\( (a) \int_2^8 \frac{dx}{x^2 - 4} \)
\( (b) \int_0^1 \frac{dx}{x^3 + x^5} \)
Hint 1: We have the p-test for improper integrals of the second kind. The proof is exactly
like that of the first kind, namely, we apply the limit definition.
Here is the statement of the p-test for improper integrals of the second kind.
\( \int_0^b \frac{dx}{x^p} = \begin{cases} \text{divergent,} & \text{if } p \ge 1\\ \frac{b^{-p+1}}{-p+1}, & \text{if } p < 1. \end{cases} \)
Hint 2: For part (a), use substitution \( u = x - 2 \).

1. Use the Comparison Test to determine whether the following integrals converge, diverge or
the test is inconclusive.
(a) ∫2^8 (dx)/(x^2 - 4)
(b) ∫0^1 (dx)/(x^3 + x^5)
Hint 1: We have the p-test for improper integrals of the second kind. The proof is exactly
like that of the first kind, namely, we apply the limit definition.
Here is the statement of the p-test for improper integrals of the second kind.
∫0^b (dx)/(x^p) = divergent, if p ≥ 1
(b^-p+1)/(-p+1), if p < 1.
Hint 2: For part (a), use substitution u = x - 2.

Added by Jose Maria T.

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