If ₀^ 2x^3 - 3 x^2 + kx 1+ x dx = (12 - ^2) , then the value of k is

Step-by-step solution

Let I = ₀^ 2x^3 - 3 x^2 + kx 1+ x dx Using the property ₀^ a f(x) dx = ₀^ a f(a-x) dx , we replace x with - x : I = ₀^ 2( - x)^3 - 3 ( - x)^2 + k( - x) 1+ ( - x) dx I = ₀^ 2( ^3 - 3 ^2 x + 3 x^2 - x^3) - 3 ( ^2 - 2 x + x^2) + k - kx 1+ x dx I = ₀^ -2x^3 + 3 x^2 - kx + k - ^3 1+ x dx Adding the two expressions for I : 2I = ₀^ k - ^3 1+ x dx 2I = (k - ^3) ₀^ 1 1+ x dx To evaluate the integral, we use the property ₀^ 2a f(x) dx = 2 ₀^ a f(x) dx if f(2a-x) = f(x) : ₀^ 1 1+ x dx = 2 ₀^ /2 1 1+ x dx = 2 ₀^ /2 1 1+ ( /2 -