Find the definition field of function y = √ SiNx / 1 + cosx

Find the definition field of function y = √ SiNx / 1 + cosx

The domain must meet the following requirements:
sinx>=0--> 2kπ=

The known function y = SiNx + cosx gives the following four propositions ① If x ∈ [0, π / 2], y ∈ (0, √ 2] ② straight line x = π / 4 is a symmetry axis of the image of function y = SiNx + cosx ③ on the interval [π / 4,5 π / 4], function y = SiNx + cosx is an increasing function ④ the image of function y = SiNx + cosx can be obtained by shifting π / 4 units from the image of y = √ 2sinx to the right, and the sequence number of the correct proposition is?

y=sinx+cosx
=√2((√2/2)sinx+(√2/2)cosx)
=√2sin(x+π/4)
Therefore, the image of this function is to shift y = √ 2sinx to the left π / 4
So wrong
① When x ∈ [0, π / 2], it is exactly a section where y = √ 2sinx is in [π / 4,3 π / 4], y ∈ (2, √ 2), so it is incorrect
② If the ordinate axis is translated to x = π / 4, the function image is consistent with y = √ 2cosx, so x = π / 4 is the axis of symmetry. Therefore, this article is correct
Similarly, on the interval [π / 4,5 π / 4], the function y = SiNx + cosx is a subtractive function, so ③ is incorrect

Given the square of the function y = (SiNx + cosx) + the square of 2cosx, Find: 1. Decreasing interval 2. Maximum value, minimum value and the value range of X when obtaining the maximum and minimum value

y=(sinx+cosx)^2+2(cosx)^2
=(sinx)^2+2sinxcosx+(cosx)^2+2(cosx)^2
=1+sin2x+2(cosx)^2
=2+sin2x+2(cosx)^2-1
=2+sin2x+cos2x
=2+√2(√2/2sin2x+√2/2cos2x)
=2+√2(sin2xcosπ/4+sinπ/4cos2x)
=2+√2sin(2x+π/4)
1. The decreasing interval is: [K π + π / 8, K π + 5 π / 8], and K is an integer
2. The maximum value is 2 + √ 2. When x = k π + π / 8, the maximum value is obtained
The minimum value is 2 - √ 2. When x = k π + 5 π / 8, the minimum value is obtained

The minimum positive period of the function y = SiNx + cosx is __

∵y=sinx+cosx═
2sin（x+π
4），∴T=2π
1=2π．
So the answer is 2 π

Calculate the indefinite integral ∫ (2-xsinx) / X DX Calculate the indefinite integral ∫ (2-xsinx) / X DX

∫(2-xsinx)/x dx
=∫(2/x-sinx) dx
=2lnx+cosx+C

Solving indefinite integral: ∫ x ^ 2 / (xsinx + cosx) ^ 2 DX I can't solve it for a long time,

Just tried with Mathlab, its indefinite integral can't be expressed by elementary function, it belongs to transcendental integral, so don't think about it anymore
The following is the operation result of Mathlab:
>> F=int('x^2/(x*sin(x)+cos(x))^2')
F1=simplify(F)\x0b
pretty(F1)
Warning: explicit integral could not be found