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Question

Question: \(\int_{}^{}\sqrt{1 + \cos ecx}\)dx =...

1+cosecx\int_{}^{}\sqrt{1 + \cos ecx}dx =

A

± sin–1 (tan x – sec x) + c

B

2sin–1 (cos x) + c

C

sin–1(cosx2sinx2)\left( \cos\frac{x}{2} - \sin\frac{x}{2} \right) + c

D

± 2 sin–1(sinx2cosx2)\left( \sin\frac{x}{2} - \cos\frac{x}{2} \right) + c

Answer

± 2 sin–1(sinx2cosx2)\left( \sin\frac{x}{2} - \cos\frac{x}{2} \right) + c

Explanation

Solution

I = dx

I = dx = ± sinx2+cosx22sinx2cosx2\int_{}^{}\frac{\sin\frac{x}{2} + \cos\frac{x}{2}}{\sqrt{2\sin\frac{x}{2}\cos\frac{x}{2}}}dx

I = ± sinx2+cosx21(sinx2cosx2)2\int_{}^{}\frac{\sin\frac{x}{2} + \cos\frac{x}{2}}{\sqrt{1 - \left( \sin\frac{x}{2} - \cos\frac{x}{2} \right)^{2}}}dx

Put sin x2\frac{x}{2} – cos x2\frac{x}{2} = t, (cosx2+sinx2)\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right) dx = 2dt

I = ± 2dt1t2\int_{}^{}\frac{2dt}{\sqrt{1 - t^{2}}} = ± 2 sin–1(t)

= ± 2 sin–1 (sinx2cosx2)\left( \sin\frac{x}{2} - \cos\frac{x}{2} \right) + c