A) \[\sin ({{e}^{{{x}^{3}}}})+c\]
B) \[\frac{1}{3}\sin ({{e}^{{{x}^{3}}}})+c\]
C) \[-\frac{1}{3}\sin ({{e}^{{{x}^{3}}}})+c\]
D) \[3\sin ({{e}^{{{x}^{3}}}})+c\]
Correct Answer: B
Solution :
Let \[I=\int{{{x}^{2}}{{e}^{2}}}\cos ({{e}^{{{x}^{3}}}})dx\] Put \[{{e}^{{{x}^{3}}}}=t\] \[\Rightarrow \] \[{{e}^{{{x}^{3}}}}.3{{x}^{2}}dx=dt\] \[\Rightarrow \] \[{{x}^{2}}{{e}^{2}}dx=\frac{1}{3}dt\] \[\therefore \] \[I=\int{\frac{1}{3}\cos t\,dt}\] \[=\frac{\sin t}{3}+c\] \[=\frac{\sin ({{e}^{{{x}^{3}}}})}{3}+c\]
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