Evaluate : (a) `int_(u)^(upsilon)dv = int_(0)^(t)adt
(b)int_(0)^(s)ds=int_(0)^(t)udt+int_(0)^(t)adt
(c) int_(u)^(upsilong)udv=int_(0)^(s)ads`

Answer

(a) `upsilon = u+at` (b) `s = ut +(1)/(2)at^2` (c ) `upsilon^2 = u^2 +2 as`

Explanation

(a) `overset(upsilon)undersetuint = int_0^t a dt or oversetupsilonundersetu = a(t)_0^t or upsilon - u =at`
(b) `overset(upsilon)undersetuintds=oversettunderset0intat dt`
`(s)_0^s = u(t)_0^t +a ((t^2)/(2))_0^t or (s-0) = u (t-0)+(1)/(2)a (t^2 -0^2) or s = ut +(1)/(2)at ^2`
(c)`int_u^(upsilon) upsilon d upsilon = int_0^s ads or ((upsilon^2)/(2))_u ^(upsilon) = a(s)_0^s`
or `(upsilon^2)/(2) - (u^2)/(2)= as or upsilon^2 = u^2 +2 as`