Question

Etermine if the given improper integrals converge or diverge:

(a) ∫[0 to [infinity]] 5/√[4(1+x)] dx
(b) ∫[-[infinity] to 1] 1/(x²+9) dx;
(c) ∫[-[infinity] to [infinity]]"

Answer 1

To analyze whether the improper integrals converge or diverge, evaluate each integral individually by either direct calculation or by comparison with known convergent or divergent integrals.

Step-by-step explanation:

To determine if the given improper integrals converge or diverge, we need to analyze each one individually:

Improper Integral (a)

The integral ∫[0 to ∞] 5/√[4(1+x)] dx is an improper integral because it has an infinite limit of integration. We can rewrite the integral as ∫[0 to ∞] ⅓ dx/√(1+x) and then determine its convergence by comparing it to a known convergent integral or by evaluating it directly if possible.

Improper Integral (b)

The integral ∫[-∞ to 1] 1/(x²+9) dx is also an improper integral due to the infinite limit. This integral can be evaluated using trigonometric substitution or by recognizing it as the integral of a function that behaves similarly to 1/x² for large values of x, which is known to converge.

For both integrals, if we can find that the integral converges to a finite value, then we say the improper integral converges; if not, it diverges.