Definite Integral Evaluation

Question

∫³₀ x² Sqrt(x+1) dx ?

Answer 1

Let u = x+1   Then, x = u-1 and dx = du

 

                     When x = 0, u = 1 and when x = 3, u = 4

 

So, the given integral can be expressed as ∫_{(1 to 4)} (u-1)²√udu

 

            = ∫_{(1 to 4)} (u²-2u+1)u^½du

 

            = ∫_{(1 to 4)} (u^5/2 - 2u^3/2 + u^1/2)du

 

            = [(2/7)u^7/2 - (4/5)u^5/2 + (2/3)u^3/2)_{(from 1 to 4)}

 

            = [256/7 - 128/5 + 16/3] - [2/7 - 4/5 + 2/3]

 

            = 254/7 - 124/5 + 14/3 = 1696/105≈16.1524