The value of the integral ₀^ _e(x) x^2 + 4 ,dx is:

Step-by-step solution

Let I = ₀^ _e(x) x^2 + 4 ,dx Substitute x = 2 , which gives dx = 2 ^2 ,d . The limits of integration change from x = 0 = 0 to x = 2 . I = ₀^ /2 _e(2 ) 4 ^2 + 4 (2 ^2 ) ,d I = ₀^ /2 _e(2 ) 4 ^2 (2 ^2 ) ,d I = 1 2 ₀^ /2 _e(2 ) ,d I = 1 2 ₀^ /2 ( _e 2 + _e( )) ,d I = 1 2 _e 2 ₀^ /2 1 ,d + 1 2 ₀^ /2 _e( ) ,d Let I₁ = ₀^ /2 _e( ) ,d Using the property ₀^a f(x) ,dx = ₀^a f(a-x) ,dx , we get: I₁ = ₀^ /2 _e ( ( 2 - ) ) ,d = ₀^ /2 _e( ) ,d I₁ = ₀^ /2 _e ( 1 ) ,d = - ₀^ /2 _e( ) ,d = -I₁ 2I₁ = 0 I₁ = 0 Substituting I₁ = 0 ba