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Let f(x) = vmatrix x^2 & x & (x^2+1) x & x^3 & x x & x & 2x vmatrix . If I = _ - /4 ^ /4 ( x f(x) x^4+1 + 24 ^2 x 1+e^x ) dx , then the value of I is equal to _____.

Correct answer

24

Step-by-step solution

Let f(x) = vmatrix x^2 & x & (x^2+1) x & x^3 & x x & x & 2x vmatrix . Replacing x with -x , we get: f(-x) = vmatrix (-x)^2 & (-x) & ((-x)^2+1) (-x) & (-x)^3 & -x (-x) & -x & (-2x) vmatrix f(-x) = vmatrix x^2 & x & (x^2+1) - x & -x^3 & -x - x & -x & - 2x vmatrix Taking -1 common from the second and third rows: f(-x) = (-1)(-1) vmatrix x^2 & x & (x^2+1) x & x^3 & x x & x & 2x vmatrix = f(x) Thus, f(x) is an even function. This implies that x f(x) x^4+1 is an odd function. The integral of an odd function over a symmet

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