Mathematics

If $$f\left( x \right) =\left| \begin{matrix} x & \cos { x }  & { e }^{ { x }^{ 2 } } \\ \sin { x }  & { x }^{ 2 } & \sec { x }  \\ \tan { x }  & { x }^{ 4 } & { 2x }^{ 2 } \end{matrix} \right| $$ then $$\int _{ -\pi /2 }^{ \pi /2 }{ f(x) } dx=$$


ANSWER

$$0$$


SOLUTION
$$\begin{array}{l} Let\, \, \, f\left( x \right) \, =\left| { \begin{array} { { 20 }{ c } }x & { \cos  x } & { { e^{ { x^{ 2 } } } } } \\ { \sin  x } & { { x^{ 2 } } } & { \sec  x } \\ { \tan  x } & { { x^{ 4 } } } & { 2{ x^{ 2 } } } \end{array} } \right|  \\ f\left( { -x } \right) \, =\left| { \begin{array} { { 20 }{ c } }{ -x } & { \cos  (-x) } & { { e^{ { { (-x) }^{ 2 } } } } } \\ { \sin  (-x) } & { { { (-x) }^{ 2 } } } & { \sec  (-x) } \\ { \tan  (-x) } & { { { (-x) }^{ 4 } } } & { 2{ { (-x) }^{ 2 } } } \end{array} } \right|  \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\left| { \begin{array} { { 20 }{ c } }{ -x } & { \cos  x } & { { e^{ { x^{ 2 } } } } } \\ { -\sin  x } & { { x^{ 2 } } } & { \sec  x } \\ { -\tan  x } & { { x^{ 4 } } } & { 2{ x^{ 2 } } } \end{array} } \right|  \\ taking\, out\, common\, -1\, from\, { c_{ 1 } } \\ =-1\left| { \begin{array} { { 20 }{ c } }x & { \cos  x } & { { e^{ { x^{ 2 } } } } } \\ { \sin  x } & { { x^{ 2 } } } & { \sec  x } \\ { \tan  x } & { { x^{ 4 } } } & { 2{ x^{ 2 } } } \end{array} } \right|  \\ =-f\left( x \right)  \\ So\, ,\, f\left( x \right) \, is\, an\, odd\, funcyion\,  \\ therefore\, \, ,\, \int _{ -\frac { \pi  }{ 2 }  }^{ \frac { \pi  }{ 2 }  }{ f\left( x \right) dx=0 }  \end{array}$$
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Single Correct Medium Published on 17th 09, 2020
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