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Lai
  • Member for 4 years, 6 months
  • Last seen this week
  • Hong Kong
19 votes
7 answers
2k views

Can the integral be found without Feynman’s trick?

18 votes
7 answers
890 views

Is there any method other than Feynman’s trick which can deal further with powers higher than 2?

15 votes
5 answers
1k views

How far can we go with $\int_{0}^{\infty} \frac{\ln \left(1+x^n\right)}{1+x^{2}} dx \textrm{ where }n\in N ?$

14 votes
3 answers
518 views

Is there any other simpler method to evaluate $\int_0^{\infty} \frac{x^n-2 x+1}{x^{2 n}-1} d x,$ where $n\geq 2?$

14 votes
7 answers
1k views

How to find exact value of integral $\int_{0}^{\infty} \frac{1}{\left(x^{4}-x^{2}+1\right)^{n}}dx$?

13 votes
3 answers
793 views

How do we find the exact value of $\int_{0}^{\infty} \frac{\ln ^{n}\left(1+x^{2}\right)}{1+x^{2}} d x$, where $n\in \mathbb N?$

13 votes
1 answer
355 views

Is there a simpler method to compute $\int_0^{\infty} \frac{1}{(1+x)\left(\pi+\ln ^{2n } x\right)} d x$

12 votes
7 answers
627 views

Can we compute $\int_1^{\infty} \frac{\sin ^2(\ln x)}{x^2 \ln ^2 x} d x$ without Feynman’s trick?

12 votes
5 answers
595 views

How far can we go with the integral $I_n=\int_0^1 \frac{\ln \left(1-x^n\right)}{1+x^2} d x$

12 votes
1 answer
438 views

Can we evaluate $\int \frac{1}{\sin ^ {2n+1} x+\cos ^ {2n+1} x} d x?$

12 votes
4 answers
365 views

How to find the exact value of the integral $ \int_{0}^{\infty} \frac{d x}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}}$?

11 votes
5 answers
498 views

Seeking help to find the exact value of $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\sin ^{2n} x+\cos ^{2n} x} d x $ using substitutions?

10 votes
1 answer
175 views

How can I find the formula for the integral $ \int_{0}^{\infty} \frac{x^{n}\left(e^{3 x}-e^{x}\right)}{\left(e^{x}-1\right)^{4}} d x ? $ where $n>2$.

9 votes
1 answer
343 views

Can we prove that $\int_0^{2\pi} e^{r \cos \theta} \cos (r \sin \theta+n \theta) d \theta=0$ without complex analysis?

8 votes
3 answers
415 views

Are there any other decent methods to evaluate $\int_0^1 \frac{\ln \left(1-x^4\right)}{1+x^2} d x?$

8 votes
0 answers
238 views

How to find $ \int_0^1 \frac{x^n}{(1-x) \ln ^n(1-x)} d x? $

8 votes
5 answers
282 views

How to generalise the limit of the product $\lim _{n \rightarrow \infty} \prod_{k=1}^n\left(1+\frac{1}{n}+\frac{k}{n^2}\right) $?

8 votes
2 answers
344 views

A contour integration approach for $\int_{0}^{\infty} \frac{\ln \left(x^{4}+2 x^{2} \cos 2 a+1\right)}{1+x^{2}} \, d x$

8 votes
1 answer
356 views

Is there any closed form for the integral $\int_0^{\frac{\pi}{2}} \frac{\ln ^n(\sin x) \ln ^m(\cos x)}{\tan x} d x?$

8 votes
4 answers
334 views

Interesting integral $\int_{0}^{\frac{\pi}{2}} \frac{d x}{\left(1+\sin ^{2} x\right)^{2}}$

8 votes
5 answers
287 views

Is there any other method to show that $\int_{0}^{\frac{\pi}{2}} x \ln (\sin x) d x =-\frac{\pi^{2}}{8} \ln 2+\frac{7}{16}\zeta(3)?$

8 votes
3 answers
274 views

Can we evaluate the integral $ I(a)=\int_0^{\infty} \frac{\sin x}{x^a} e^{-x} d x, $ without Gamma functions?

7 votes
4 answers
590 views

How to find $\int_{0}^{\pi} \ln (b \cos x+c)$ without using Feynman’s integration technique?

7 votes
0 answers
175 views

What happens when an even natural number $n$ meets the integral $\int_{0}^{\frac{\pi}{2}} x\tan 2 x \ln^n (\tan x) d x$?

7 votes
3 answers
287 views

Do we have a simpler method for computing $\int_{-\infty}^{\infty} \frac{\ln \left(x^2+ax+b\right)}{1+x^2} d x$, where $b> \frac{a^2}{4} $?

7 votes
3 answers
264 views

How to deal with odd $m$ in integral $\int_{0}^{\frac{\pi}{4}}(\sin^{6}m x+\cos^{6}m x) \ln (1+\tan x) d x $

7 votes
2 answers
150 views

How many method to evaluate the integral $\int_{0}^{1} \frac{\ln ^{n}(1-x)}{x} d x , \textrm{ where }n\in N?$

7 votes
1 answer
317 views

Is there any formula for $S(k)=\sum_{n=1}^{\infty} \frac{1}{n^{k} k^{n}},$ where $k\in \mathbb{N}$?

7 votes
2 answers
330 views

How far can I go with the integral $\int \frac{\sin ^{n} x \cos ^{n} x}{1-\sin x \cos x} d x, $ where $n\in N$?

7 votes
1 answer
240 views

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 $p \in N$?

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