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$\int \cos e c^{2} \sqrt{2^{x}} 2^{x} d x is equal to$

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a

−2log22xcot2x+logsin2x+C

b

2log22xcot2x−logsin2x+C

c

2log2logsin2x−2xcot2x+C

d

2log2logsin2x+2xcot2x+C

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detailed solution

Correct option is C

Let I=∫cos⁡ec22x2xdx Put 2x=y⇒122x2xlog⁡2dx=dy⇒2xdx=2ylog⁡2dyI=2log⁡2∫cos⁡ec2yydy=2log⁡2y∫cosec2⁡y−∫dydy∫cos⁡ec2ydydy=2log⁡2−ycot⁡y+∫cot⁡ydy=2log⁡2[−ycot⁡y+log⁡sin⁡y]+C=2log⁡2log⁡sin⁡2x−2xcot⁡2x+C

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