User user170231

14 votes

What is an intuitive approach to solving $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$?

answered Jul 27, 2017 at 16:49

12 votes

Accepted

How to evaluate $\int \frac{1+x}{1+\sqrt x} dx$

answered Aug 15, 2016 at 16:07

10 votes

Accepted

Find the value of the integral: $\int_{0}^{1}\frac{x}{1+x^4}\arctan(x)\,\mathrm{d}x$

answered Nov 1 at 21:24

9 votes

Any different way to solve an integral involving a trigonometric ratio?

answered Jul 26 at 19:16

9 votes

Accepted

Prove $_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)=\frac{27}{16}\left(\pi^2-7\zeta(3)\right) $

answered Aug 21 at 17:25

9 votes

Is Wolfram Alpha linear independence wrong or am I missing something?

answered Aug 11, 2016 at 15:41

8 votes

Accepted

A log log integral

answered Nov 28, 2022 at 7:30

8 votes

Accepted

Finding $x-\frac{1}{x}$, given $x^3 - \frac{1}{x^3} = 108+76\sqrt{2}$

answered Oct 18, 2022 at 17:11

8 votes

Accepted

Area of loop in graph

answered Jul 3 at 18:04

8 votes

Accepted

Is it necessary to write limits for a substituted integral?

answered Feb 19, 2020 at 16:27

7 votes

Evaluate $\int_0^1\arcsin^2(\frac{\sqrt{-x}}{2}) (\log^3 x) (\frac{8}{1+x}+\frac{1}{x}) \, dx$

answered Dec 12, 2022 at 8:42

7 votes

Accepted

Proving $\sum_{n=0}^{\infty}\frac{1}{2^{n}}\sum_{r=0}^{n}\frac{1}{2^{4r}}\binom{2r}{r}^2=\frac{1}{\pi\sqrt{\pi}}\Gamma\left(\frac{1}{4}\right)^2$

answered Nov 3 at 17:22

7 votes

How to integrate ratio of summations $\int_0^1 \sum_{i=0}^{2n} x^{i} /\sum_{i=0}^{n} x^{i}\ dx$

answered Oct 17 at 18:37

7 votes

Accepted

Calculate $\displaystyle \lim_{x \to 3} \frac{\sqrt{19-x} - 2\sqrt[4]{13+x}}{\sqrt[3]{11-x} - x + 1}$

answered Jan 29, 2021 at 20:09

6 votes

Accepted

Show that $\prod_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$

answered Nov 13, 2022 at 7:29

6 votes

Accepted

$\int_0^\infty\frac{\ln{(x^8+x^4+2x^2+1)}}{x^2+1}dx$ and $\int_0^\infty\frac{1}{x^2+1}\arctan{\left(\frac{x^2+1}{x^4}\right)}dx$

answered Nov 16, 2022 at 3:27

6 votes

Accepted

How to Evaluate $\int_0^1 {\frac{{u\arcsin u}}{{{u^4} + 2{u^2} + 13}}du} .$?

answered Dec 4 at 19:25

6 votes

How to evaluate $\int_0^1 \frac{dx}{\sqrt{x + \sqrt{x^2 + \sqrt{x^3}}}}?$

answered Jul 18 at 6:44

6 votes

evaluation of $\int_0^\infty \frac{\ln^2(x)}{1+x^2}dx$

answered Apr 26 at 16:44

6 votes

Accepted

Using u-substitution on an integral

answered Dec 6, 2020 at 23:26

6 votes

Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$

answered Feb 12, 2016 at 23:42

5 votes

Accepted

Spot the mistake $\vert e^{iz}\vert = \sqrt{cos^2(z)+sin^2(z)}$

answered Dec 13, 2022 at 22:09

5 votes

When $\cos(x)+\cos(y)=\sin(x)+\sin(y)=1$, $x$ or $y$ must be a multiple of $2\pi$?

answered Apr 10 at 5:47

5 votes

Integral : from Summation to Integral notation

answered Mar 27 at 21:25

5 votes

Accepted

Simplifying Elliptic Integrals

answered May 6 at 5:55

5 votes

How might I figure out the $n$th term, and the general Maclaurin/Taylor series, for $\sin^3 x$?

answered May 7 at 7:19

5 votes

Tricky trig integral

answered Jul 27 at 22:02

5 votes

What is the sum of the following infinite series?

answered Mar 17, 2020 at 19:46

5 votes

Accepted

The integral $\int\frac{{\rm d}x}{(x+3)^{\frac87}(x-2)^{\frac 67}}$

answered Jan 26, 2016 at 20:25

5 votes

Accepted

I want to solve $\int \frac{2}{x^2(x^2+1)^2}dx$

answered Feb 3, 2016 at 18:05

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666 Answers

What is an intuitive approach to solving $\lim_{n\rightarrow\infty}\biggl(\frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2}+\dots+\frac{n}{n^2}\biggr)$?

How to evaluate $\int \frac{1+x}{1+\sqrt x} dx$

Find the value of the integral: $\int_{0}^{1}\frac{x}{1+x^4}\arctan(x)\,\mathrm{d}x$

Any different way to solve an integral involving a trigonometric ratio?

Prove $_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)=\frac{27}{16}\left(\pi^2-7\zeta(3)\right) $

Is Wolfram Alpha linear independence wrong or am I missing something?

A log log integral

Finding $x-\frac{1}{x}$, given $x^3 - \frac{1}{x^3} = 108+76\sqrt{2}$

Area of loop in graph

Is it necessary to write limits for a substituted integral?

Evaluate $\int_0^1\arcsin^2(\frac{\sqrt{-x}}{2}) (\log^3 x) (\frac{8}{1+x}+\frac{1}{x}) \, dx$

Proving $\sum_{n=0}^{\infty}\frac{1}{2^{n}}\sum_{r=0}^{n}\frac{1}{2^{4r}}\binom{2r}{r}^2=\frac{1}{\pi\sqrt{\pi}}\Gamma\left(\frac{1}{4}\right)^2$

How to integrate ratio of summations $\int_0^1 \sum_{i=0}^{2n} x^{i} /\sum_{i=0}^{n} x^{i}\ dx$

Calculate $\displaystyle \lim_{x \to 3} \frac{\sqrt{19-x} - 2\sqrt[4]{13+x}}{\sqrt[3]{11-x} - x + 1}$

Show that $\prod_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$

$\int_0^\infty\frac{\ln{(x^8+x^4+2x^2+1)}}{x^2+1}dx$ and $\int_0^\infty\frac{1}{x^2+1}\arctan{\left(\frac{x^2+1}{x^4}\right)}dx$

How to Evaluate $\int_0^1 {\frac{{u\arcsin u}}{{{u^4} + 2{u^2} + 13}}du} .$?

How to evaluate $\int_0^1 \frac{dx}{\sqrt{x + \sqrt{x^2 + \sqrt{x^3}}}}?$

evaluation of $\int_0^\infty \frac{\ln^2(x)}{1+x^2}dx$

Using u-substitution on an integral

Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$

Spot the mistake $\vert e^{iz}\vert = \sqrt{cos^2(z)+sin^2(z)}$

When $\cos(x)+\cos(y)=\sin(x)+\sin(y)=1$, $x$ or $y$ must be a multiple of $2\pi$?

Integral : from Summation to Integral notation

Simplifying Elliptic Integrals

How might I figure out the $n$th term, and the general Maclaurin/Taylor series, for $\sin^3 x$?

Tricky trig integral

What is the sum of the following infinite series?

The integral $\int\frac{{\rm d}x}{(x+3)^{\frac87}(x-2)^{\frac 67}}$

I want to solve $\int \frac{2}{x^2(x^2+1)^2}dx$