father of all integrturzzzz>>>>>

first write everything in terms of `x/2` [submultiple angles].then divide throughout by cos⁴x/2.then take tanx/2=t.thats is. (that is cosx/2 to the power of 4.)

well....

consider the denominator...

sin^2x+sinx+1... break it into two complex factors...

sin^2x+sinx+1 = [sinx + (1/2 + i√3/2)][sinx + (1/2 - i√3/2)]

now,,,

I = ∫dx/[sinx + (1/2 + i√3/2)][sinx + (1/2 - i√3/2)]

do partials....

I = (1/i√3) ∫[sinx + (1/2 + i√3/2) - sinx - (1/2 - i√3/2)] dx/ [sinx + (1/2 + i√3/2)][sinx + (1/2 - i√3/2)]

I= (1/i√3) ∫dx/[sinx + (1/2 - i√3/2)] - (1/i√3) ∫dx/[sinx + (1/2 + i√3/2)]

now solvable :)

if 0 ∫ Λ/2 sinx = k(means area of the curve from 0 to Λ/2 uia is k)

Whats the value of

a 0∫Λ/4 (1 +tanx)