Question

If f(x) = cosx – cos2x + cos3x – cos4x + ____ then ∫f(x)dx = ____ + c

(a) tan(x/2)

(b) x + tan(x/2)

(b) x – (1/2)tan(x/2)

(d) x – tan(x/2)

Answer 1

The correct option (d) x – tan(x/2)

Explanation:

f(x) = cosx – cos2x + cos3x – cos4x + ……..

here a = cosx

r = – cosx

limn➙∞ sn = [a/(1 – r)]

∴ f(x) = [(cosx)/(1 + cosx)]

∴ ∫f(x)dx = ∫[(cosx)/(1 + cosx)]dx

= ∫[{cos(1 – cosx)}/{(1 + cosx) (1 – cosx)}]dx

= ∫[(cosx – cos2x)/(1 – cos2x)]dx

= [{cosx – (1– sin2x)}/(sin2x)]dx

= ∫[(cosx)/(sin2x)]dx – ∫[1/(sin2x)]dx + ∫1 ∙ dx

= – [1/(sinx)] + cotx + x + c

= [(cosx – 1)/(sinx)] + x + c

= [{– 2sin2(x/2)}/{2sin(x/2) ∙ cos(x/2)}] + x + c

= x – tan(x/2) + c