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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
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Solution
Let `I = int_0^2 x sqrt (2 - x) dx`
Put 2 - x = t
⇒ dx = dt
When x = 0, t = 2
and x = 2, t = 0
∵ `I = - int_2^0 (2 - t) sqrtt dt`
`= int_0^2 (2t^(1/2) - t^(3/2)) dt`
`= [(2t^(3/2))/(3/2) - t^(5/2)/(5/2)]_0^2` `...[∵ - int_a^0 f (x) dx = int_0^a f (x) dx]`
`= [4/3 t^(3/2) - 2/5 t^(5/2)]_0^2`
`= 4/3 (2)^(3/2) - 2/5 (2)^(5/2)`
`= 4/3 xx 2 sqrt2 - 2/5 xx 4 sqrt2`
`= (8sqrt2)/3 - (8 sqrt 2)/5`
`= (16 sqrt2)/15`
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