Question

Solve the simultaneous equation:
$\frac{1}{3x} - \frac{1}{4y} + 1 = 0$
$\frac{1}{5x} + \frac{1}{2y} = \frac{4}{15}$
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Answer 1

Mark as BRAINLIEST answer

Answer:

OLD ANSWER

Proposition. If 0<r<1 and r<s, then

I(r,s):=∫1−11x1+x1−x−−−−−√log(1+2rsx+(r2+s2−1)x21−2rsx+(r2+s2−1)x2)dx=4πarcsinr.(1)

Assuming this proposition, all that we have to do is to solve the non-linear system of equations

2rs=2andr2+s2−1=2.

The unique solution satisfying the condition of the proposition is r=ϕ−1 and s=ϕ. So by (1) we have

∫1−11x1+x1−x−−−−−√log(1+2x+2x21−2x+2x2)dx=I(ϕ−1,ϕ)=4πarcsin(ϕ−1)=4πarccotϕ−−√.

Thus it remains to prove the proposition.