A) \[{{e}^{x}}-{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c\]
B) \[{{e}^{-x}}-{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c\]
C) \[{{e}^{x}}+{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c\]
D) \[{{e}^{x}}-{{e}^{\sin y}}-\frac{{{x}^{3}}}{3}=c\]
E) \[{{e}^{x}}-{{e}^{\sin y}}+\frac{{{x}^{3}}}{3}=c\]
Correct Answer: C
Solution :
Given, \[\cos y\frac{dy}{dx}={{e}^{x+\sin y}}+{{x}^{2}}{{e}^{\sin y}}\] \[\Rightarrow \] \[\cos y\frac{dy}{dx}={{e}^{\sin y}}({{e}^{x}}+{{x}^{2}})dx\] \[\Rightarrow \] \[\int{\frac{\cos y}{{{e}^{\sin y}}}dy}=\int{({{e}^{x}}+{{x}^{2}})dx}\] Put \[\sin y=t\]in \[LHS\Rightarrow \cos ydy=dt\] \[\therefore \] \[\int{\frac{dt}{{{e}^{t}}}}=\int{({{e}^{x}}+{{x}^{2}})}dx\] \[\Rightarrow \] \[-{{e}^{-t}}={{e}^{x}}+\frac{{{x}^{3}}}{3}-c\] \[\Rightarrow \] \[{{e}^{x}}+{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c\]
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