Let f(x) = vmatrix 128( 2+ 4+x^4 - 2) & (1+2x^4) e^ 3x^4 - 1 & 1+2x^4 - 1 vmatrix . The value of _ x 0 f(x) x^8 is :

Step-by-step solution

We need to evaluate _ x 0 f(x) x^8 . Dividing the determinant by x^8 is equivalent to dividing each element of the 2 2 matrix by x^4 . _ x 0 f(x) x^8 = vmatrix _ x 0 128( 2+ 4+x^4 - 2) x^4 & _ x 0 (1+2x^4) x^4 _ x 0 e^ 3x^4 - 1 x^4 & _ x 0 1+2x^4 - 1 x^4 vmatrix Let us evaluate each limit separately: L₁ = _ x 0 128( 2+ 4+x^4 - 2) x^4 Rationalizing the numerator: L₁ = _ x 0 128(2+ 4+x^4 - 4) x^4( 2+ 4+x^4 + 2) = _ x 0 128( 4+x^4 - 2) x^4(2+2) = _ x 0 32( 4+x^4 - 2) x^4 Rationalizing again: L₁ = _ x 0 32(4+x^4 - 4) x