The correct option (b) (1/2)(sinx – cosx) – (x/2)
Explanation:
I = ∫[1/(tanx + cotx + secx + cosecx)] ∙ dx
= ∫[1/({(sinx + 1)/(cosx)} + {(cosx + 1)/(sinx)})]dx
= ∫[{1 (sinx ∙ cosx)}/(sin2x + sinx + cos2x + cosx)]dx
= ∫[(sinx ∙ cosx)/(1 + sinx + cosx)]dx
= ∫[{sinx cosx(sinx + cosx – 1)}/{(sinx + cosx + 1)(sinx + cosx – 1)}]dx
= ∫[{sinx cosx(sinx + cosx – 1)}/{(sinx + cosx)2 – 1}]dx
= ∫[{sinx cosx(sinx + cosx – 1)}/(2sinx cosx)]dx
= (1/2)[∫sinxdx + ∫cosxdx – ∫1 ∙ dx] = (1/2)(sinx – cosx) – (x/2) + c