Evaluate interal from 1 to 5 of x^2sqrt (x-1)dx. Thanks.

Question

Show steps. Thanks.

Answer 1

∫_{(from 1 to 5)} x²√(x-1)dx   Let u = x-1   Then x = u+1 and dx = du

 

                                           When x = 1, u = 1-1 = 0

                                           When x = 5, u = 5-1 = 4

 

= ∫_{(from 0 to 4)} (u+1)² √u du

 

= ∫_{(from 0 to 4)} [(u²+2u+1)√u]du

 

=∫_{(from 0 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 0 to 4)} 

 

= 256/7 + 128/5 + 16/3

 

= 7088/105 ≈ 67.5