A) \[2({{x}^{2}}-{{y}^{2}})=3x\]
B) \[2({{x}^{2}}-{{y}^{2}})=6y\]
C) \[x({{x}^{2}}-{{y}^{2}})=6\]
D) \[x({{x}^{2}}+{{y}^{2}})=10\]
Correct Answer: A
Solution :
Let \[I=\int{\frac{{{e}^{x}}\,dx}{\sqrt{5-4{{e}^{x}}-{{e}^{2x}}}}}\] Put\[{{e}^{x}}=t\] \[\Rightarrow \] \[{{e}^{x}}dx=dt\] \[\therefore \] \[I=\int{\frac{dt}{\sqrt{5-4t-{{t}^{2}}}}}\] \[=\int{\frac{dt}{\sqrt{5({{t}^{2}}+4t+4)+4}}}\] \[I=\int{\frac{dt}{\sqrt{9-{{(t+2)}^{2}}}}}\] \[\Rightarrow \] \[I={{\sin }^{-1}}\left( \frac{t+2}{3} \right)+c\] \[\Rightarrow \] \[I={{\sin }^{-1}}\left( \frac{{{e}^{x}}+2}{3} \right)+c\]
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