`int_(-1)^1 t(1 - t)^2 dt` Evaluate the integral

Textbook Question

Chapter 5, 5.4 - Problem 26 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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