Question

Upper limit is x/2 lower limit is 0 Integrate √cotxintegrate√cotx / √cotx√tanx =π/4 /√cotx+√tanx = π/4

Answer 1

Answer:

0∫π/2 tanx/(tanx + secx).dx

= 0∫π/2 tanx*(secx - tanx)/(tanx + secx)*(secx - tanx).dx

= 0∫π/2 tanx*(secx - tanx)/(sec^2 x - tan^2 x) dx

= 0∫π/2 tanx*(secx - tanx) dx

= 0∫π/2 (tanx sec x - tan^2 x) dx

= 0∫π/2 (tanx sec x - sec^2 x + 1) dx

= 0∫π/2 [tan(π/2 - x) sec (π/2 - x) - sec^2 (π/2 - x) + 1] dx

= 0∫π/2 (cotx cscx - csc^2 x +1) dx

= [-csc x + cotx + x] (x=0 to π/2)

=[ - csc(π/2) + cot(π/2) + π/2 ] - lim x -> 0 [ - cscx + cotx]

= [ -1 + π/2 ] - lim x-> 0 [(cosx - 1)/sinx]

= π/2 - 1 - lim x->0 [ -sinx / cosx ] [L'Hospital's Theorem]

= π/2 - 1 - 0

= π/2 - 1.

Step-by-step explanation: