A) \[{{a}^{x}}{{e}^{x}}+c\]
B) \[\left[ \frac{{{a}^{x}}{{e}^{x}}}{\log a} \right]+c\]
C) \[\left[ \frac{{{(ae)}^{x}}}{(x+1)} \right]+c\]
D) \[\left[ \frac{{{a}^{x}}{{e}^{x}}}{1+\log a} \right]+c\]
E) \[\frac{{{e}^{x}}{{a}^{x}}}{(a+1)}+c\]
Correct Answer: D
Solution :
Let \[I=\int{{{(ae)}^{x}}}dx=\int{{{a}^{x}}{{e}^{x}}}dx\] \[\Rightarrow \] \[I=\int{{{a}^{x}}{{e}^{x}}}dx\] \[=\frac{{{a}^{x}}}{\log a}{{e}^{x}}-\int{{{e}^{x}}\frac{{{a}^{x}}}{\log a}}dx\] \[\Rightarrow \] \[I=\frac{{{a}^{x}}{{e}^{x}}}{\log a}-\frac{I}{\log a}\] \[\Rightarrow \] \[I+\frac{I}{\log a}=\frac{{{a}^{x}}{{e}^{x}}}{\log a}\] \[\Rightarrow \] \[(\log a+1)I={{a}^{x}}{{e}^{x}}\] \[\Rightarrow \] \[I=\frac{{{a}^{x}}{{e}^{x}}}{\log a+1}+c\]
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