# Evaluate $int_0^{pi/2} frac{arctan{left(frac{2sin{x}}{2cos{x}-1}right)}sin{left(frac{x}{2}right)}}{sqrt{cos{x}}} , mathrm{d}x$

Mathematics Asked by user801111 on September 23, 2020

Evaluate: $$int_0^{frac{pi}{2}} frac{arctan{left(frac{2sin{x}}{2cos{x}-1}right)}sin{left(frac{x}{2}right)}}{sqrt{cos{x}}} , mathrm{d}x$$

I believe there is a "nice" closed form solution but Wolfram is too weak. These arctan integrals are so tricky! I sense a substitution like $$sin{frac{x}{2}}$$ because of arctan argument and $$sqrt{cos{x}}$$ but I just cant get it. Any ideas or tips please.

Source: https://tieba.baidu.com/p/4794735082 (Exercise 3.1.22).

$$boxed{I=int_0^frac{pi}{2}arctanleft(frac{2sin x}{2cos x -1}right)frac{sinleft(frac{x}{2}right)}{sqrt{cos x}}dx=sqrt 2 pi lnvarphi-frac{pi}{sqrt 2}ln(2+sqrt 3), varphi =frac{1+sqrt 5}{2}}$$ To show this result, we'll do some substitutions until it's clear how to simplify that $$arctan$$ term. $$I ,overset{cos x=t}=frac1{sqrt 2}int_0^1 arctan left(frac{2sqrt{1-t^2}}{2t-1}right)frac{dt}{sqrt tsqrt{1+t}}overset{large t=frac{1-x}{1+x}}=int_0^1 frac{arctanleft(frac{4sqrt x}{1-3x}right)}{sqrt{1-x}(1+x)}dx$$ $$=int_0^1frac{arctan (sqrt x)+arctan(3sqrt x)}{sqrt{1-x}(1+x)}dx-int_frac13^1frac{pi}{sqrt{1-x}(1+x)}dx=mathcal J-frac{pi}{sqrt2}ln(2+sqrt 3)$$ The second integral appears since for $$x>frac13$$ we have $$sqrt x cdot 3sqrt x>1$$.
Now in order to show $$mathcal J=sqrt 2piln varphi$$ we can differentiate under the integral sign, considering: $$mathcal J(a)=int_0^1frac{arctan(asqrt x)+arctan(3sqrt x)}{sqrt{1-x}(1+x)}dx$$ $$Rightarrow mathcal J'(a)=int_0^1 frac{sqrt x}{sqrt{1-x}(1+x)(1+a^2 x)}dxoverset{frac{1-x}{x}=t}=int_0^infty frac{dt}{sqrt t(2+t)(1+a^2+t)}$$ $$overset{t=x^2}=frac{2}{1-a^2}int_0^infty left(frac{1}{1+a^2+x^2}-frac{1}{2+x^2}right)dx=frac{pi}{1-a^2}left(frac{1}{sqrt{1+a^2}}-frac{1}{sqrt 2}right)$$ We are looking to find $$mathcal J(1)=mathcal J$$, but $$mathcal J(-3)=0$$ therefore our integral is: $$mathcal J=int_{-3}^1 frac{pi}{1-a^2}left(frac{1}{sqrt{1+a^2}}-frac{1}{sqrt 2}right)daoverset{-a=frac{1-x}{1+x}}=frac{pi}{2sqrt 2}int_0^2frac{1}{x}left(1-frac{1-x}{sqrt{1+x^2}}right)dx$$ $$=frac{pi}{2sqrt 2}left(lnleft(x+sqrt{1+x^2}right)+lnleft(1+sqrt{1+x^2}right)right)bigg|_0^2=sqrt 2 pi ln varphi$$

Correct answer by Zacky on September 23, 2020

## Related Questions

### (proposed) elegant solution to IMO 2003 P1

0  Asked on January 18, 2021 by mnishaurya

### Reverse order of polynomial coefficients of type $left(r-xright)^n$

1  Asked on January 18, 2021 by thinkingeye

### Should one include already cemented proofs of related principles in one’s paper?

0  Asked on January 18, 2021 by a-kvle

### The Cauchy-Crofton formula on a plane

1  Asked on January 18, 2021 by jalede-jale-uff-ne-jale

### Solving a limit for capacity of a transmission system

3  Asked on January 18, 2021 by jeongbyulji

### Galois group Abstract algebra

1  Asked on January 18, 2021 by user462999

### Does there always exists coefficients $c,dinmathbb{R}$ s.t. $ax^3+bx^2+cx+d$ has three different real roots?

2  Asked on January 17, 2021 by w2s

### If there is a “worldly ordinal,” then must there be a worldly cardinal?

2  Asked on January 17, 2021 by jesse-elliott

### Cover number and matching number in hypergraphs.

1  Asked on January 17, 2021 by josh-ng

### $displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ in an ellipse

1  Asked on January 17, 2021 by fabrizio-gambeln

### Show that $h_n(x)=x^{1+frac{1}{2n-1}}$ converges uniformly on $[-1, 1]$.

1  Asked on January 17, 2021 by wiza

### What does the functional monotone class theorem say and how does it relate to the other monotone class theorem?

0  Asked on January 17, 2021 by alan-simonin

### Symmetric matrix and Hermitian matrix, unitarily diagonalizable

2  Asked on January 17, 2021

### direct conversion from az/el to ecliptic coordinates

0  Asked on January 16, 2021 by klapaucius-klapaucius

### A proof problem in mathematical statistics

2  Asked on January 16, 2021

### How to find the z component of the parameterization of an ellipse that is the intersection of a vertical cylinder and a plane

1  Asked on January 16, 2021 by nono4271

### using Lagrange multipliers to determine shortest distance between a point and straight line

2  Asked on January 16, 2021 by am_11235

### Proof of Lemma 5.1.5.3 in Jacob Lurie’s HTT.

1  Asked on January 16, 2021 by robin-carlier