Let I ₁ = ₀^ 4 ^2 x 1+ x x d x and I ₂ = ₀^ 4 ^2 x 1+ x x d x . The value of 12 e^ I ₂ - I ₁ + 9 3 ( I ₁ + I ₂) is equal to ________.

Step-by-step solution

First, consider the sum of the two integrals: I ₁ + I ₂ = ₀^ 4 ^2 x + ^2 x 1+ x x d x = ₀^ 4 1 1+ x x d x Divide the numerator and the denominator by ^2 x : I ₁ + I ₂ = ₀^ 4 ^2 x ^2 x + x d x = ₀^ 4 ^2 x 1 + ^2 x + x d x Let t = x , so d t = ^2 x d x . The limits change from 0 1 : I ₁ + I ₂ = ₀^1 d t t^2 + t + 1 = ₀^1 d t (t + 1 2 )^2 + 3 4 I ₁ + I ₂ = 2 3 [ ⁻¹ ( 2t+1 3 ) ]₀^1 = 2 3 [ ⁻¹ ( 3 ) - ⁻¹ ( 1 3 ) ] I ₁ + I ₂ = 2 3 ( 3 - 6 ) = 3 3 Next, consider the difference of the two integrals: I ₂ - I ₁ = ₀^ 4 ^2 x -