Question

\(\mathop \smallint \nolimits_{\frac{{\rm{\pi }}}{4}}^{\frac{{\rm{\pi }}}{4}} \frac{{\cos \left( {{{\rm{e}}^{3{\rm{x}}}}} \right)}}{{{{\rm{x}}^4} + {{\rm{x}}^3} + 1}} =\)
1. ¼
2. 1
3. 0
4. ½
5. None of these

Answer 1

Correct Answer - Option 3 : 0

Concept:

• \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = 0\)

Calculation:

We know, \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = 0\)

Here, limit of integration is same (i.e., π/4)

\(\therefore \mathop \smallint \nolimits_{\frac{{\rm{\pi }}}{4}}^{\frac{{\rm{\pi }}}{4}} \frac{{\cos \left( {{{\rm{e}}^{3{\rm{x}}}}} \right)}}{{{{\rm{x}}^4} + {{\rm{x}}^3} + 1}} = 0\)

Hence, option (3) is correct.