Problem 1079
Suppose that \(\displaystyle \int_0^{\infty} |f|\) is convergent. Does it follow that \(\displaystyle \lim_{\infty} f=0\)? Difficulty: 2. |
Problem 1086
Is the following integral convergent? \(\displaystyle \int_0^3 {\cos t \over t}\ \dt\) Difficulty: 2. |
Problem 1084
\(\displaystyle \lim_{0+0} x\cdot \int_x^1 {\cos t \over t^2} \dt =?\) Difficulty: 3. |
Problem 1063
Prove that \(\displaystyle \int_0^{\infty} x^ne^{-x}\ \dx=n! .\) Difficulty: 5. |
Problem 1080
Show that if \(\displaystyle f\) is uniformly continuous on \(\displaystyle [2, \infty)\), then \(\displaystyle \int_0^{\infty} {f(x) \over x^2\log^2 x}\ \dx\) is convergent. Difficulty: 5. |
Problem 1087
\(\displaystyle \int_0^{\pi/2} \log \cos x\ \dx=?\) Difficulty: 5. |
Problem 1088
For what \(\displaystyle \alpha\) is \(\displaystyle \int_0^1 (x-\sin x)^{\alpha}\ \dx\) convergent? Difficulty: 5. |
Problem 1062
Are the following improper integrals convergent? Absolute convergent? \(\displaystyle a) \int_1^{\infty} {\sin x\over x^2}\ \dx \qquad\qquad b)\ \int_1^{\infty} {\sin x\over x}\ \dx \qquad\qquad c)\ \int_1^{\infty}\sin (x^2)\ \dx\) Difficulty: 6. |
Problem 1089
Is there a continuous function \(\displaystyle f:\R\to\R\) for which \(\displaystyle \int_0^{\infty} f\) is convergent, but \(\displaystyle \int_0^{\infty}f^2\) is divergent? Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |