Calculus concepts：練習問題

Determine the convergence of the improper integral \( \int_{1}^{\infty} \frac{\ln x}{x^2} dx \) and find its value if it converges.

Consider the sequence of integrals \( J_n = \int_{0}^{\infty} x^n e^{-x^2} dx \) for \( n \geq 1 \). Which of the following represents \( J_n \) in terms of \( J_{n-2} \) for \( n \geq 2 \)?

Consider the integral \( I_n = \int_{0}^{1} x^n e^{-x} dx \) for \( n \geq 1 \). Which of the following recurrence relations correctly describes the relationship between \( I_n \) and \( I_{n-1} \)?

Evaluate the improper integral \( \int_{0}^{\infty} \frac{1}{(1+x)\sqrt{x}} dx \) by using the substitution \( u = \sqrt{x} \).

Let \( I_n = \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx \). Using the reduction formula \( I_n = \frac{n-1}{n} I_{n-2} \) for \( n \geq 2 \), evaluate the value of \( I_5 \).

Consider the definite integral \(I_n = \int_{0}^{1} (1-x^2)^n \, dx\) for \(n \ge 0\).

(a) By using integration by parts, derive the reduction formula: \(I_n = \frac{2n}{2n+1} I_{n-1}\) for \(n \ge 1\).
(b) Given that \(I_0 = 1\), use the result in part (a) to find an explicit expression for \(I_n\) in terms of factorials or double factorials.
(c) Determine whether the improper integral \(\int_{0}^{\infty} e^{-x^2} \, dx\) is convergent by relating it to the behavior of \(I_n\) as \(n \to \infty\).