4.34.3 Can you answer these questions?

4.34.3.1 Strange result for an improper integral $\int_ 0 ^ 1 \frac x^ -1/2   e^ x -1  \sin\left(\frac 1  x \right)dx$

https://mathematica.stackexchange.com/questions/155066/strange-result-for-an-improper-integral-int-01-fracx-1-2ex-1-s

 
The improper integral $$\int_{0}^{1}\frac{x^{-1/2}}{e^{x}-1} \sin\left(\frac{1}{x}\right)dx$$ 
is convergent  conditionally. 
 
(Applying Dirichlet Test,$$\frac{x^{-1/2}}{e^{x}-1}  ...
 

- asked by user250236 1vote

4.34.3.2 Perplexing simplification of RotationMatrix

https://mathematica.stackexchange.com/questions/154793/perplexing-simplification-of-rotationmatrix

 
The result of 
 
FullSimplify[RotationMatrix[, {Cos[], Sin[], 0}],  > 0].{0, 0, 1} 
 
 
in Mathematica 11 is 
 
{Sign[Sec[]]^2 Sin[] Sin[], -Cos[] Sin[], Cos[]} 
 
 
instead of the nicer result  ...
 

- asked by theindigamer 1vote

4.34.3.3 collecting terms with different order

https://mathematica.stackexchange.com/questions/154849/collecting-terms-with-different-order

 
I am using homotopy perturbation method to solve nonlinear ODE. 
 
The first step is to introduce P parameter and collect different order of P. 
 
The code is like this: 
 
Collect[(1 - 
p) ((((1 + 
...
 

- asked by sajad.sharhani 1vote

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