Question

Why does$ \int_{0}^{\pi} \frac{dx}{\sqrt{\sin(x)}}$ converge while $\int_{0}^{\pi} \frac{dx}{\sin(x)}$ diverges?

score 217 · Answer 1 · 2026-04-16 09:22:57

217

Views

Why does$ \int_{0}^{\pi} \frac{dx}{\sqrt{\sin(x)}}$ converge while $\int_{0}^{\pi} \frac{dx}{\sin(x)}$ diverges?

Published on 16 Apr 2026 - 9:22

score 56 · Answer 2 · 2026-04-16 08:47:21

56

Views

For which values of $r$ and $s$ does $ \int_{0}^{2} \frac{(1+x)^r-(1+x)^s}{x^2} dx $ converge?

Published on 16 Apr 2026 - 8:47

score 80 · Answer 3 · 2026-04-18 09:11:18

80

Views

Conditional convergence of $\int_0^\infty(-1)^{\lfloor x^2 \rfloor }dx$

Published on 18 Apr 2026 - 9:11

score 119 · Answer 4 · 2026-04-15 20:21:35

119

Views

$\int_{-\infty}^{\infty}e^{-x^n}dx$ converges $\iff n$ is even

Published on 15 Apr 2026 - 20:21

score 173 · Answer 5 · 2026-04-16 14:43:52

173

Views

For what values of $r$ does $\int_{1}^{\infty} \frac{cos(x)}{x^r} dx$ converge?

Published on 16 Apr 2026 - 14:43

score 156 · Answer 6 · 2026-04-16 06:15:13

156

Views

Investigating convergence of $\int_{0}^{\infty} \cos(x^r) dx $ and $\int_{\pi}^{\infty} \frac{\cos x}{\log x}\arctan\lfloor x\rfloor dx$

Published on 16 Apr 2026 - 6:15

score 60 · Answer 7 · 2026-04-16 07:46:06

60

Views

Is there a function $f : \mathbb{R}\to \mathbb{R}$ , $f>0$, $\lim_{x\to \infty}f(x) = \infty$ but $\int_{0}^{\infty}f(x)dx < \infty $?

Published on 16 Apr 2026 - 7:46

score 140 · Answer 8 · 2026-04-14 22:28:29

140

Views

Why is $\lim_{s \to 0} \int_{0}^{\infty} \frac{\sin(t)}{t}\cdot e^{-st} dt =\int_{0}^{\infty} \frac{\sin(t)}{t} dt $ legitimate?

Published on 14 Apr 2026 - 22:28

score 341 · Answer 9 · 2026-04-16 16:41:33

341

Views

How can we tackle this integral $\int_{0}^{1}{2x^2-2x+\ln[(1-x)(1+x)^3]\over x^3\sqrt{1-x^2}}\mathrm dx=-1?$

Published on 16 Apr 2026 - 16:41

score 404 · Answer 10 · 2026-03-25 23:36:50

404

Views

Integral of log and exponential with power

Published on 25 Mar 2026 - 23:36

score 67 · Answer 11 · 2026-04-16 04:01:47

67

Views

Limit $ \lim_{n\to\infty}\int^{n}_{\beta}\frac{\alpha}{u(u^{\alpha}+2+u^{-\alpha})}$

Published on 16 Apr 2026 - 4:01

score 1k · Answer 12 · 2026-04-17 10:26:39

1k

Views

Schläfli-type integral

Published on 17 Apr 2026 - 10:26

score 116 · Answer 13 · 2026-04-12 23:15:00

116

Views

Need help with $\int_{0}^\infty x^n\sin(bx)e^{-ax^2}\, dx$

Published on 12 Apr 2026 - 23:15

score 156 · Answer 14 · 2026-04-15 22:28:44

156

Views

Find $\int_0^\infty (A x +I)^{-1} B (A x +I)^{-1} dx$ where $A$ and $B$ are positive definite matrices

Published on 15 Apr 2026 - 22:28

score 49 · Answer 15 · 2026-04-16 16:52:16

49

Views

$\int_{\mathbb{R}}f(x)+g(x) \;dx<+\infty$ iff $ \int_{\mathbb{R}}g(x)\;dx<+\infty$ and $\int_{\mathbb{R}}f(x)\;dx<+\infty .$

Published on 16 Apr 2026 - 16:52

List Question

Why does$ \int_{0}^{\pi} \frac{dx}{\sqrt{\sin(x)}}$ converge while $\int_{0}^{\pi} \frac{dx}{\sin(x)}$ diverges?

For which values of $r$ and $s$ does $ \int_{0}^{2} \frac{(1+x)^r-(1+x)^s}{x^2} dx $ converge?

Conditional convergence of $\int_0^\infty(-1)^{\lfloor x^2 \rfloor }dx$

$\int_{-\infty}^{\infty}e^{-x^n}dx$ converges $\iff n$ is even

For what values of $r$ does $\int_{1}^{\infty} \frac{cos(x)}{x^r} dx$ converge?

Investigating convergence of $\int_{0}^{\infty} \cos(x^r) dx $ and $\int_{\pi}^{\infty} \frac{\cos x}{\log x}\arctan\lfloor x\rfloor dx$

Is there a function $f : \mathbb{R}\to \mathbb{R}$ , $f>0$, $\lim_{x\to \infty}f(x) = \infty$ but $\int_{0}^{\infty}f(x)dx < \infty $?

Why is $\lim_{s \to 0} \int_{0}^{\infty} \frac{\sin(t)}{t}\cdot e^{-st} dt =\int_{0}^{\infty} \frac{\sin(t)}{t} dt $ legitimate?

How can we tackle this integral $\int_{0}^{1}{2x^2-2x+\ln[(1-x)(1+x)^3]\over x^3\sqrt{1-x^2}}\mathrm dx=-1?$

Integral of log and exponential with power

Limit $ \lim_{n\to\infty}\int^{n}_{\beta}\frac{\alpha}{u(u^{\alpha}+2+u^{-\alpha})}$

Schläfli-type integral

Need help with $\int_{0}^\infty x^n\sin(bx)e^{-ax^2}\, dx$

Find $\int_0^\infty (A x +I)^{-1} B (A x +I)^{-1} dx$ where $A$ and $B$ are positive definite matrices

$\int_{\mathbb{R}}f(x)+g(x) \;dx<+\infty$ iff $ \int_{\mathbb{R}}g(x)\;dx<+\infty$ and $\int_{\mathbb{R}}f(x)\;dx<+\infty .$

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