CHAMSI's user avatar
CHAMSI's user avatar
CHAMSI's user avatar
CHAMSI
  • Member for 4 years, 1 month
  • Last seen this week
  • France
15 votes
Accepted

Prove that $\int_{0}^{\infty}\frac{(\arctan x)^3}{x^3}dx=\frac{3π}{2}\ln2-\frac{π^3}{16}$

11 votes

$\int_0^{\frac{\pi}{2}}\ln(\sin^2 x+k^2\cos^2 x)dx$ not by differentiation under the integral?

10 votes
Accepted

Convergent Improper integral whose integrand tends to a non zero finite limit as x tends to infinity.

10 votes
Accepted

Compute $\int_{0}^{\infty} \frac{\arctan{x}}{1+x} \frac{dx}{\sqrt[4]{x}}$

9 votes
Accepted

Algebraic proof that $\sum\limits_{k=0}^{n} {n \choose k}\cdot \frac{(-1)^k}{(k+1)(k+2)} = \frac{1}{n+2}$

8 votes

Evaluating $\lim_{n\to \infty} n \int_{0}^{1} \frac{x^n}{x+1} dx$

8 votes

Prove that for every $n \in \mathbb{N}$ $\sum\limits_{k=2}^{n}{\frac{1}{k^2}}<1$

7 votes
Accepted

Calculating $\lim_{x\to 0} \frac{a^{\tan x} - a^{\sin x}}{\tan x - \sin x}$ without using L'Hospital rule

7 votes

Show that $\int_0^\pi \log( 1 - 2r\cos(t) + r^2)\, dt=0$

6 votes
Accepted

Differentiability of $f(x)$ at $0$ provided that $\lim_{x\to0} [f(2x)-f(-x)]/x$ exists?

6 votes
Accepted

Calculate $\int_0^1 \left\lfloor \frac{2020}{x} \right\rfloor - 2020\left\lfloor\frac1x\right\rfloor\,dx$

6 votes

Find $\int_{0}^{\infty} \frac{\log(x) }{\sqrt{x} (x+1)^{2}}\,dx$

6 votes

Sum the series of $\sum\limits_{n=1}^{\infty} \frac{1}{2^n n} $

6 votes
Accepted

How do you solve the following sum?

6 votes
Accepted

Can one integral can give more than one answer one with natural log and other with tan inverse

6 votes

How to prove Double Factorial as Infinity Sum?

6 votes

Approximate $\sum\limits_{k=0}^{m-1}\frac{k}{m-k}$

5 votes

Calculate Indefinite Integral $\int \frac{x^5(1-x^6)}{(1+x)^{18}}dx$

5 votes
Accepted

$\int_{-\infty}^{\infty} \frac{e^{mx}}{(ae^{nx}+b)}dx$

5 votes
Accepted

Summation on floor function

5 votes
Accepted

Sum of $\sum_{n=1}^{\infty}\frac{n!}{\prod_{i=1}^{n}(x+i)}$

5 votes

Evaluating two improper integrals involving powers of $\sin(t)$

5 votes
Accepted

Evaluating $\sum_{r=1}^{3n-1}\dfrac{(-1)^{r-1}\cdot r}{\binom{3n}r}$

5 votes
Accepted

$\sum_{k=0}^n \sum_{j=0}^k { {n} \choose {j}} { {n-j} \choose {k-j}} \left( \frac{x}{1-x}\right)^k = \left( \frac{1+x}{1-x} \right)^n$

5 votes
Accepted

Calculate limit of this sequence $I_n = \int_{0}^{\pi/2} (\tan x)^{1/n} dx$

5 votes
Accepted

Evaluate $\lim\limits_{n\to \infty}\frac{1^p+3^p+\dots+(2n-1)^p}{n^{p+1}}$

5 votes
Accepted

Binomial series problem $\lim_{n\to\infty}\left[\sum_{i,j}{2i\choose i}{2j\choose j}\right]^{1/n}=4$

5 votes
Accepted

Help with $\int\frac{(x^{2}-1)}{(x^{2}+1)\sqrt{x^{4}+1}}dx$

5 votes
Accepted

Show that $\int_0^1\frac{\sqrt{\sin x}}{x}dx$ converges

5 votes

Find $\lim\limits_{x \to \infty}{\mathrm{e}^{-x}\int_{0}^{x}{f\left(y\right)\mathrm{e}^{y}\,\mathrm{d}y}}$

1
2 3 4 5
13