₀^3 ~d x (x+2) x+1 =

Step-by-step solution

Evaluate the integral ₀^3 d x (x+2) x+1 using substitution. Let u = x+1 , so u^2 = x + 1 and x = u^2 - 1 . Differentiating gives d x = 2u , d u . The limits transform from x=0 to u=1 and from x=3 to u=2 . Substituting into the integral: ₁^2 2u , d u ((u^2-1)+2)u = ₁^2 2u , d u (u^2+1)u = ₁^2 2 , d u u^2+1 The antiderivative of 1 u^2+1 is ⁻¹(u) . Therefore: 2 [ ⁻¹(u) ]₁^2 = 2 ( ⁻¹(2) - ⁻¹(1) ) Using the identity ⁻¹(A) - ⁻¹(B) = ⁻¹ ( A-B 1+AB ) with A=2 , B=1 : ⁻¹(2) - ⁻¹(1) = ⁻¹ ( 1 3 ) Thus, the integral simplifies