integration of √sin^-1(√4x-1)
Answers
Answered by
0
This requires Fresnel integral functions. It is not easy.
Let 4 x - 1 = sin² (t/2)
4 dx = 2 sin (t/2) * cos (t/2) dt / 2 = Sin t dt / 2
![I= \int {\sqrt{Sin^{-1}(\sqrt{4x-1})}} \, dx\\\\=\frac{1}{8\sqrt{2}}\int\ {\sqrt{t}*sin\ t} \, dt \\\\=\frac{1}{8\sqrt2}*[\sqrt{\frac{\pi}{2}}\ C(\sqrt{\frac{2}{\pi}}*\sqrt{t})-\sqrt{t}*Cos\ t]\\\\Here, C(x)=Fresnel\ Integral=\int\limits_0^x {cos(\frac{\pi y^2}{2})} \, dy I= \int {\sqrt{Sin^{-1}(\sqrt{4x-1})}} \, dx\\\\=\frac{1}{8\sqrt{2}}\int\ {\sqrt{t}*sin\ t} \, dt \\\\=\frac{1}{8\sqrt2}*[\sqrt{\frac{\pi}{2}}\ C(\sqrt{\frac{2}{\pi}}*\sqrt{t})-\sqrt{t}*Cos\ t]\\\\Here, C(x)=Fresnel\ Integral=\int\limits_0^x {cos(\frac{\pi y^2}{2})} \, dy](https://tex.z-dn.net/?f=I%3D+%5Cint+%7B%5Csqrt%7BSin%5E%7B-1%7D%28%5Csqrt%7B4x-1%7D%29%7D%7D+%5C%2C+dx%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B8%5Csqrt%7B2%7D%7D%5Cint%5C+%7B%5Csqrt%7Bt%7D%2Asin%5C+t%7D+%5C%2C+dt+%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B8%5Csqrt2%7D%2A%5B%5Csqrt%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5C+C%28%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%7D%7D%2A%5Csqrt%7Bt%7D%29-%5Csqrt%7Bt%7D%2ACos%5C+t%5D%5C%5C%5C%5CHere%2C+C%28x%29%3DFresnel%5C+Integral%3D%5Cint%5Climits_0%5Ex+%7Bcos%28%5Cfrac%7B%5Cpi+y%5E2%7D%7B2%7D%29%7D+%5C%2C+dy)
Let 4 x - 1 = sin² (t/2)
4 dx = 2 sin (t/2) * cos (t/2) dt / 2 = Sin t dt / 2
kvnmurty:
click on red heart thanks button above pls
Similar questions
Math,
10 months ago
Computer Science,
10 months ago
Math,
10 months ago
Math,
1 year ago
Social Sciences,
1 year ago
English,
1 year ago