Question

How to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0\Rightarrow\lim\limits_{x\rightarrow\infty}f'(x)$?

score 82 · Answer 1 · 2026-04-16 05:15:34

How to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0\Rightarrow\lim\limits_{x\rightarrow\infty}f'(x)$?

score 921 · Answer 2 · 2026-04-12 07:10:35

921

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Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

Published on 12 Apr 2026 - 7:10

score 69 · Answer 3 · 2026-04-13 06:52:41

69

Views

Correctness of the integral $\int_a^bf(x)\,dx$ when $f$ is not defined at $a$ and $b$

Published on 13 Apr 2026 - 6:52

score 105 · Answer 4 · 2026-04-11 06:52:34

105

Views

Computation of integral

Published on 11 Apr 2026 - 6:52

score 90 · Answer 5 · 2026-04-17 07:47:12

90

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Proving $\int_{0}^{\infty} e^{-x^2}dx=\sqrt{n}\int_{0}^{\infty} e^{-nx^2}dx$

Published on 17 Apr 2026 - 7:47

score 234 · Answer 6 · 2015-11-19 05:25:31

234

Views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

Published on 19 Nov 2015 - 5:25

score 75 · Answer 7 · 2026-04-16 09:33:13

75

Views

Does $\int^3_0 (9-x^2)^{-3/2}\, dx$ converge?

Published on 16 Apr 2026 - 9:33

score 208 · Answer 8 · 2026-04-17 13:49:03

208

Views

Evaluation of $ \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}dx$

Published on 17 Apr 2026 - 13:49

score 548 · Answer 9 · 2026-04-16 00:30:11

548

Views

Find $k$ such that $\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$

Published on 16 Apr 2026 - 0:30

score 60 · Answer 10 · 2026-04-11 06:42:18

60

Views

General technique to check the convergence of an improper integral?

Published on 11 Apr 2026 - 6:42

score 185 · Answer 11 · 2026-04-05 15:15:43

185

Views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

Published on 05 Apr 2026 - 15:15

score 281 · Answer 12 · 2026-04-15 03:48:36

281

Views

Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $

Published on 15 Apr 2026 - 3:48

score 120 · Answer 13 · 2026-04-17 15:21:31

120

Views

Test the convergence of the integral $\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$

Published on 17 Apr 2026 - 15:21

score 95 · Answer 14 · 2026-04-16 20:19:12

95

Views

How to compute $\int_{-\infty}^{+\infty} 2^{-4^t}(1-2^{-4^t})\,dt=\frac{1}{2}.$

Published on 16 Apr 2026 - 20:19

score 37 · Answer 15 · 2026-04-13 01:07:22

37

Views

Integral evaluation by sandwiching

Published on 13 Apr 2026 - 1:07

List Question

How to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0\Rightarrow\lim\limits_{x\rightarrow\infty}f'(x)$?

Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

Correctness of the integral $\int_a^bf(x)\,dx$ when $f$ is not defined at $a$ and $b$

Computation of integral

Proving $\int_{0}^{\infty} e^{-x^2}dx=\sqrt{n}\int_{0}^{\infty} e^{-nx^2}dx$

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

Does $\int^3_0 (9-x^2)^{-3/2}\, dx$ converge?

Evaluation of $ \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}dx$

Find $k$ such that $\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$

General technique to check the convergence of an improper integral?

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $

Test the convergence of the integral $\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$

How to compute $\int_{-\infty}^{+\infty} 2^{-4^t}(1-2^{-4^t})\,dt=\frac{1}{2}.$

Integral evaluation by sandwiching

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