Calculus: Early Transcendentals, Chapter 5, 5.4, Section 5.4, Problem 26

You need to use the following substitution to evaluate the definite integral, such that:
1 - t = u => -dt = du
int_(-1)^1 t*(1-t)^2dt = int_(u_1)^(u_2) (1 - u)*u^2 (-du)
int_(u_1)^(u_2) (u - 1)*u^2 (du) = int_(u_1)^(u_2)u^3 (du) - int_(u_1)^(u_2)u^2 (du)
int_(u_1)^(u_2) (u - 1)*u^2 (du) = (u^4/4 - u^3/3)|_(u_1)^(u_2)
int_(-1)^1 t*(1-t)^2dt = (((1-t)^4)/4 - ((1-t)^3)/3)|_(-1)^1
int_(-1)^1 t*(1-t)^2dt = (((1-1)^4)/4 - ((1-1)^3)/3) - (((1+ 1)^4)/4 - ((1+1)^3)/3)
int_(-1)^1 t*(1-t)^2dt = 8/3 - 16/4 = 8/3 - 4 = -4/3
Hence, evaluating the definite integral yields int_(-1)^1 t*(1-t)^2dt = -4/3.