A) \[\frac{1}{4}\ln \left( \frac{2-\sin x+\cos x}{2+\sin x+\cos x} \right)+c\]
B) \[\frac{1}{2}\ln \]\[\left( \frac{2+\sin x}{2-\sin x} \right)+c\]
C) \[\frac{1}{4}\ln \]\[\left( \frac{1+\sin x}{1-\sin x} \right)+c\]
D) none of the above
Correct Answer: A
Solution :
Let \[I=\int_{{}}^{{}}{\frac{\sin x+\cos x}{3+\sin 2x}}dx\] \[=\int_{{}}^{{}}{\frac{\sin x+\cos x}{4+\sin 2x-1}}dx\] \[\Rightarrow \] \[I=\int_{{}}^{{}}{\frac{\sin x+\cos x}{4-{{(\sin x-\cos x)}^{2}}}}dx\] Let \[\sin x-\cos x=t\] \[(\cos x+\sin x)dx=dt\] \[\therefore \] \[I=\int_{{}}^{{}}{\frac{dt}{4-{{t}^{2}}}=\frac{1}{4}\log }\frac{2-t}{2+t}+c\] \[=\frac{1}{4}\log \frac{2-(\sin x-\cos x)}{2+(\sin x+\cos x)}+c\] \[=\frac{1}{4}\log \left( \frac{2-\sin x+\cos x}{2+\sin x+\cos x} \right)+c\]
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