(1)
let
y=t^2
dy =2tdt
t=0, y=0
t=âx, y=x
â«(0->âx ) t.sin(t^2) dt
=(1/2)â«(0->x ) siny dy
=(1/2) ( cosx -1 )
(2)
â«(0->Ï/2 ) (cosx)^5. sinx dx
=-â«(0->Ï/2 ) (cosx)^5. dcosx
=-(1/6)[ (cosx)^6] |(0->Ï/2 )
=1/6
(3)
let
y=x-1
â«(0->2 ) |x-1| dx
=â«(-1->1 ) |y| dy
=-â«(-1->0 ) y dy +â«(0->1 ) y dy
=- (1/2) [y^2]|(-1->0) + (1/2) [y^2]|(0->1)
=1
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