Correct Answer - Option 2 : -8
Concept:
Integral properties:
\(\mathop \smallint \limits_{ - {\rm{a}}}^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{\;dx}} = \left\{ {\begin{array}{*{20}{c}} {2\mathop \smallint \limits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}},If\;f\left( { - {\rm{x}}} \right) = f\left( x \right)}\\ {0,\;If\;f\left( { - {\rm{x}}} \right) = - f\left( x \right)} \end{array}} \right.\)
Calculation:
Let \({\rm{I}} = \mathop \smallint \limits_{ - 2}^2 \left\{ {4{x^3} + 2x - 2} \right\}dx\)
\(\Rightarrow {\rm{I}} = \mathop \smallint \limits_{ - 2}^2 \left( {4{x^3} + 2x} \right)dx - \mathop \smallint \limits_{ - 2}^2 2dx\)
Let I = I1 – I2
Now, \({{\rm{I}}_1} = \mathop \smallint \limits_{ - 2}^2 \left( {4{x^3} + 2x} \right)dx\)
Let’s find if the function is even or odd.
f(x) = 4x3 + 2x
Put x = -x
⇒ f(-x) = 4(-x)3 + 2(-x)
⇒ f(-x) = -(4x3 + 2x) ⇒ f(-x) = - f(x)
Hence, the given function is odd.
∴ I1 = 0
Now, \({{\rm{I}}_2} = \mathop \smallint \limits_{ - 2}^2 2dx\) = \(\left[ {2x} \right]_{ - 2}^2\) = 2 (2 – (-2)) = 2 × 4 = 8
∴ I = I1 – I2 = 0 – 8 = - 8