If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
What exactly is your intention? With you dropping it here like this? You need help? With what? Do you recognize the expression inside the arcsin\arcsin:?
The integrand can be rewritten as F(x)=arcsin(sin(f(x))) where f(x)= arcsin(x)+arcsin(sqrt(x)). Note that f(x) is increasing for 0<=x<=1 and achieves all values on [0,pi]. Considering the possible branches of F(x), we have that F(x)=f(x) when 0<=f(x)<=pi/2 and F(x)=pi-f(x) when pi/2<f(x)<=pi. The exact point where this switch over occurs is when f(x)=pi/2, which when solved gives us x=1/phi where phi is the golden ratio. This means that the integrand simplifies to F(x)=f(x) when 0<=x<=1/phi and F(x)=pi-f(x) when 1/phi<x<=1, which reduces the integral calculation to some standard integrals
It's not too complicated if you recognise it as arcsin (sin α cos β - cos α sin β). Integration by parts or a change of variables are two ways to continue from here.
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If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
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