question_answer

Evaluate \[\int_{-\pi /6}^{\pi /6}{{{x}^{3}}\,{{\cos }^{2}}x\,dx.}\]

Answer:

Let \[l=\int_{-\pi /6}^{\pi /6}{{{x}^{3}}{{\cos }^{2}}x\,dx}\] and       \[f(x)={{x}^{3}}{{\cos }^{2}}x\] Clearly, \[=f(-\,x)=(-\,{{x}^{3}}){{\cos }^{2}}(-\,x)\]             \[=-{{x}^{3}}{{\cos }^{2}}x\]             \[=-f(x)\] \[\therefore \] f(x) is odd function. \[\therefore \]      \[I=0\]