For first part use $\tan(a+b)= \dfrac{\tan a+\tan b}{1-\tan a \tan b}$ Put $a=x$ , $b=2x$ Further evaluate $\tan 2x$ using addition property described above by substituting $a=x$ and $b=x$ For second part use $\tan 3x$ and convert $\cot 2x$ into $\dfrac{1}{\tan 2x}$ Using the second result ...

Let's first try to imagine the quarter of the solid sphere: It has a curved surface and $2$ mutually perpendicular plane surfaces $$\text{T.S.A} = \underbrace{\frac{1}{4} 4\pi r^2}_{\text{curved surface area}} + \overbrace{\frac{1}{2}\pi r^2 + \frac{1}{2}\pi r^2}^{\text{Plane surface surface}...

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy functions: $$\operatorname{Ai}\,x=\frac{1}{\pi}\int_0^\infty\cos\left(x\,z+\frac{z^3}{3}\right)\,\mathr...

I remember that some time ago I was asking this question Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$ , and now, while I was making a review, I asked myself if we can get the closed form of $$\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$$ by using the similar tools as in that proof. The probl...