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प्रश्न
Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`
उत्तर
We use the property,
`int_0^a f(t)*dt = int_0^a f(a - t)*dt`
∴ `int_0^1 t^2 sqrtt(1 - t)*dt = int_0^1 (1 - t)^2 sqrt(1 - 1 + t)*dt`
= `int_0^1 (1 - 2t + t^2)sqrt(t)*dt`
= `int_0^1 (t^(1/2) - 2t^(3/2) + t^(5/2))*dt`
= `[(t^(3/2))/(3/2) - 2*(t(5)/(2))/(5/2) + (t^(7/2))/(7/2)]_0^1`
= `(2)/(3)(1)^(3/2) - (4)/(5)(1)^(5/2) + (2)/(7)(1)^(7/2) - 0`
= `(2)/(3) - (4)/(5) + (2)/(7) - 0`
= `(70 - 84 + 30)/(105)`
= `(16)/(105)`.
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