₀^ 4 ( x + x ) d x=

Step-by-step solution

Evaluate the integral: I = ₀^ 4 ( x + x ) , d x Begin by simplifying the integrand: x + x = x x + x x = x + x x x The integral becomes: I = ₀^ 4 x + x x x , d x Multiply numerator and denominator by 2 and recognize that 2 x x = 2x : I = ₀^ 4 2 ( x + x) 2x , d x Use the substitution t = x - x , giving d t = ( x + x) , d x . Then t^2 = 1 - 2 x x , so x x = 1 - t^2 2 . When x = 0 , t = -1 ; when x = 4 , t = 0 . Substituting: I = _ -1 ⁰ d t (1 - t^2)/2 = 2 _ -1 ⁰ 1 1 - t^2 , d t The antiderivative is t , so: I = 2 [ t]