Question

∫√(1 + secxdx) = _________ + c 

(a) – 2sin^–1(2cosx + 1) 

(b) – sin^–1(2cosx – 1) 

(c) sin^–1(2cosx – 1) 

(d) cos^–1(2cosx – 1)

Answer 1

The correct option (b) – sin^–1(2cosx – 1)   

Explanation:

consider I = ∫√(1 + secx)dx 

= ∫√[(1 + cosx)/(cosx)]dx 

= ∫√[(1 – cos²x)/(1 – cosx)(cosx)]dx 

= ∫[(sinx)/{√(cosx – cos²x)}dx] 

Let cosx = t 

∴ – sinxdx = dt 

∴ I = (– 1) × ∫[dt/√(t – t²)] 

= (– 1) × ∫[dt/√{– (t² – t)}] 

= (– 1) × ∫[dt/√{(1/4) – [t² – t + (1/4)]}] 

= (– 1) ∫[dt/√{(1/2)² – [t – (1/2)]²}] 

= (– 1) ∙ sin^–1[{t – (1/2)}/(1/2)] 

= (– 1)sin^–1(2t – 1) 

= (– 1) sin^–1(2cosx – 1)