A) \[x{{e}^{x}}+c\]
B) \[\cos (x{{e}^{x}})+c\]
C) \[\tan (x{{e}^{x}})+c\]
D) \[x\cos ec(x{{e}^{x}})+c\] Where c is a constant of integration.
Correct Answer: C
Solution :
[c] Let \[I=\int{\frac{{{e}^{x}}(1+x)}{{{\cos }^{2}}\left( x{{e}^{x}} \right)}dx}\] Put, \[x{{e}^{x}}=t\Rightarrow {{e}^{x}}(1+x)dx=dt\] \[\therefore I=\int{\frac{dt}{{{\cos }^{2}}t}=\int{{{\sec }^{2}}tdt=\tan t+c}}\] \[=\tan (x{{e}^{x}})+c\] where ?c? is a constant of integration.
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