_ - 4 ^ 4 ( x)⁻⁴ ~d x has the value

Step-by-step solution

Let aligned & I = _ - 4 ^ 4 ( x)⁻⁴ ~d x &=2 ₀^ 4 ( x)⁻⁴ ~d x & [ ( x)⁻⁴ is an even function ] &=2 ₀^ 4 1 ^4 x ~d x &=2 ₀^ 4 ^4 x ^4 x ~d x &=2 ₀^ 4 (1+ ^2 x ) ^2 x ^4 x ~d x aligned Let x= t when x=0 , we get t =0 when x= 4 , we get t =1 Also, ^2 x ~d x= dt aligned I & =2 ₀^1 1+ t ^2 t ^4 dt & =2 [ ₀^1 1 t ^4 dt + ₀^1 1 t ^2 dt ] & =2 [ [ t ⁻³ ]₀^1 -3 + [ t ⁻¹ ]₀^1 -1 ] & = -8 3 aligned