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Please solve this............. the correct answer is pi/8.
Question is attatched.
5 months ago

this ques is an application of the property that integral of f(x) from a to b is equal to integral of f(a+b – x) from a to b.
so let I= int 0_pi/2 s^3*c/(s^4+c^4) dx where s= sinx and c=cosx
now I= int 0_pi/2 c^3*s/(c^4+s^4) dx where we have replaced x by 0+pi/2 – x
now add the two eqns
2I= int 0_pi/2 sc*(s^2+c^2)/(s^4+c^4) dx=int 0_pi/2 sc/(s^4+c^4) dx
now put y=s so that dy= c dx
2I= int 0_1 y/(y^4+(1 – y^2)^2) dy
now put y^2= z so that 2ydy= dz
hence 4I= int 0_1 dz/(z^2+(1 – z)^2) dz=  int 0_1 dz/(2z^2 – 2z+1) dz
now solving this is beyond easy as this is a standard form (1/quadratic). so you can do it to get the final answer as pi/8
5 months ago
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