A) \[\log \left| \cos (x{{e}^{x}}) \right|+C\]
B) \[\log \left| \cot (x{{e}^{x}}) \right|+C\]
C) \[\log \left| sec(x{{e}^{-x}}) \right|+C\]
D) \[\log \left| sec(x{{e}^{x}}) \right|+C\]
Correct Answer: D
Solution :
[d] Let \[I=\int{\frac{(1+x){{e}^{x}}}{\cot (x{{e}^{x}})}}dx\] Put \[x{{e}^{x}}=t\Rightarrow (x{{e}^{x}}+{{e}^{x}})dx=dt\] \[\Rightarrow {{e}^{x}}(x+1)dx=dt\] \[\therefore \,\,\,\,\,I=\int{\frac{dt}{\cot (t)}=\log \left| \sec t \right|+C}\] \[=\log \left| \sec (x{{e}^{x}}) \right|+C\]
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