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plz evaluate the given integral solve each step and clear

plz evaluate the given integral solve each step and clear

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Grade:12th pass

1 Answers

Aditya Gupta
2080 Points
2 years ago
let I= int 0_2pi 1/(a+bsinx)^2 dx
f(x) can be replaced by f(a+b-x)
so I= int 0_2pi 1/(a+bsin(2pi-x))^2 dx= int 0_2pi 1/(a – bsinx)^2 dx
adding we get 2I= int 0_2pi 2(a^2+b^2sin^2x)/(a^2 – b^2sin^2x)^2 dx
or I=  a^2*int 0_2pi 1/(a^2 – b^2sin^2x)^2 dx+ b^2*int 0_2pi sin^2x/(a^2 – b^2sin^2x)^2 dx
now both the above integrals are of the standard form and can be easily solved by multiplying the numerator and denominator by sec^4x and then substituting y= tanx. i know this ques is super complicated but thats the way it is :(
           an easier approach can be to write sinx in terms of tanx/2 and then substituting u= tanx/2
kindly approve :)

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