A) \[x\tan x/2+c\]
B) \[\cot x/2+c\]
C) \[\log (1+\cos x)+c\]
D) \[\log (x+\sin x)+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{x+\sin x}{1+\cos x}\,}dx\] \[\Rightarrow \]\[\int_{{}}^{{}}{\frac{x}{1+\cos x}}dx+\int_{{}}^{{}}{\frac{\sin x}{1+\cos x}}dx\] \[\Rightarrow \]\[\int_{{}}^{{}}{\frac{x}{1+2{{\cos }^{2}}x/2-1}}+\int_{{}}^{{}}{\frac{2\sin x/2\cos x/2}{1+2{{\cos }^{2}}x/2-1}}\] \[\Rightarrow \]\[\int_{{}}^{{}}{\frac{x}{2}{{\sec }^{2}}x/2+\int_{{}}^{{}}{\tan x/2dx}}\] \[=\int_{{}}^{{}}{x.\frac{1}{2}}{{\sec }^{2}}x/2dx+\int_{{}}^{{}}{\tan x/2}\] \[=x\tan x/2-\int_{{}}^{{}}{\tan x/2}dx\] \[=x\tan x/2+c\]
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