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Lai's user avatar
Lai's user avatar
Lai
  • Member for 4 years, 6 months
  • Last seen this week
  • Hong Kong
13 votes
Accepted

How do I find $S(p)=1+\frac{x^{p}}{p !}+\frac{x^{2 p}}{(2 p) !}+\frac{x^{3 p}}{(3 p) !}+\cdots$ by complex numbers, where $p \in N$?

13 votes

How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\cos(x)}{(1 + \sqrt{\sin(2x)})^n} \,dx$

10 votes

Why is $\int_0^\infty(\frac{1}{1+x^a}-\frac{1}{1+x^b})\frac{dx}{x}=0$ for all a,b>0?

9 votes

How to tackle the integral $\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x$?

9 votes
Accepted

Is there an elegant method to find the minimum value of $\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}$ for positive $x$?

9 votes
Accepted

Prove that $\int_0^{\infty} \frac{{\rm Li}_2(-x)}{1+x^2}\, {\rm d}x = - \frac{7 \pi^3}{96}$

9 votes

How to evaluate $\int_0^1 \ln ^3(1+x) \ln (1-x) d x$?

7 votes

How to evaluate the integral $\int_0^t\frac{xe^x}{1+e^x}\,dx$?

7 votes

Evaluate $\int_{0}^{1} \frac{\operatorname{arsinh}x}{x\sqrt{1-x^4}} \,\mathrm{d}x$

7 votes

2023 MIT Integration Bee Semifinals #1 Problem 1 $\int e^{\cos x}\cos(2x+\sin x)dx$

6 votes

Prove $ \int_0^{\infty} \frac{\ln ^3 x}{(1+x)^{n}} d x =A+B \pi^2$ for some rational numbers $A$ and $B$, where $n\neq 1 $

6 votes

How can we evaluate the integral $\int_{0}^{\infty} \frac{d x}{1+x+x^{2}+\ldots+x^{n-1}}$, where $n\geq 3$?

6 votes

Evaluate: $\int_{-1}^{1}\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}dt$

6 votes

Calculus: Integral of a function

5 votes

If $\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n}$, then $n=$

5 votes

Extra solutions when solving $\sin\theta+\cos\theta=\sqrt{2\sin2\theta}$

5 votes

Solving $\int_0^{\infty} \frac{\sin^m(x)}{x^n} dx$ for $m, n \in \mathbb{Z}^+$

5 votes

Primality testing with binomial coefficients

5 votes
Accepted

Evaluate the integral $I=\int_0^1 \sqrt{-1+\sqrt{\frac{4}{x}-3}}\ dx$

5 votes

The calculation does not use complex variable function theory $\int _0^{2\pi }e^{\cos \left(x\right)}\cos \left(\sin \left(x\right)\right)dx$

5 votes
Accepted

hard integral: $\sin(x)/\sqrt{1+2\sin(2x)}$ between $0$ and $\pi/2$

5 votes

Proof that $\int_0^1\frac{\sin\ln x}{\ln x}dx=0$ is wrong

5 votes
Accepted

How do we split this integral?

5 votes

Find $\lfloor {\alpha}^6 \rfloor$

5 votes
Accepted

Challenging series evaluation

5 votes

Formula for integral of $x^a \ln^n(x) dx$

5 votes

Compute the integral: $\int_{0}^{\frac{\pi}{2}}\ln(\sec(x)+\tan(x))\csc(x)dx$

5 votes

Compute the integral: $\int_{0}^{\frac{\pi}{2}}\ln(\sec(x)+\tan(x))\csc(x)dx$

5 votes

Any cool ways to solve $\int_{0}^{\frac{\pi}{2}}\log(1+4\sin^2x)\,dx$

5 votes

How to calculate this improper integral?

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