A) \[\frac{1}{a}{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\]
B) \[a{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\]
C) \[{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\]
D) \[\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{{{e}^{x}}\left[ {{\sin }^{-1}}\frac{x}{a}+\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right]}\,dx\] \[=\int_{{}}^{{}}{{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}\,dx}+\int_{{}}^{{}}{\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}\,dx\] \[={{e}^{x}}{{\sin }^{-1}}\frac{x}{a}-\int_{{}}^{{}}{\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}\,dx+\int_{{}}^{{}}{\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}\,dx+c\] \[={{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c.\]
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