The integral\int\sqrt{1+2\cotx(\cos \ex + \cot \x) dx (where C is a constant of integration) is equal to

Correct Answer is : 2 log (sinx2) +c

NIMCET 2019 Question Paper with solutions

Question : 37 of 120

Marks: +1, -0

The integral

∫√{1+2\cotx(\cos \ex + cot \x)

dx (where C is a constant of integration) is equal to

Solution:

Let

\I=∫√{1+2\cot x({\cosec}x+\cot x)}dx

\I=∫√{1+2\cot x{\cosec}x+2\cot^{2}x }dx

\I=∫√{1+\cot^{2}x+2\cot x{\cosec}x+\cot^{2}x}dx

\I=∫√{{\cosec}^{2}x+2{\cosec}x\cot x+\cot^{2}x}dx

\I=∫√{({\cosec}x+\cot x)^{2}}dx

\I=∫|{\cosec}x+\cot x|dx

\I=∫({\cosec} x+\cot x)dx

[∵x ≤ ∊0,{π}/{2})]

\I=\log({\cosec}x-\cot x)+\log\sin x+C

\I=\log({\cosec} x-\cot x)(\sin x)+C

\I=\log(1-\cos x)+C

\I=\log 2\sin^{2}\;\;{x}/{2}+C⇒I=2\log(\sin{x}/{2})+C

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