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Question
Evaluate the following : if `int_a^a sqrt(x)*dx = 2a int_0^(pi/2) sin^3x*dx`, find the value of `int_a^(a + 1)x*dx`
Solution
It is given that
`int_a^a sqrt(x)*dx = 2a int_a^(pi/2) sin^3x*dx`
∴ `[x^(3/2)/(3/2)]_0^a = 2a*(2)/(3)` ...[Using Reduction Formula]
∴ `[(2a^(3/2))/(3) - 0] = (4a)/(3)`
∴ `(2asqrt(a))/(3) = (4a)/(3)`
∴ `2a(sqrta - 2)` = 0
∴ a = 0 or `sqrt(a)` = 2
i.e. a = 0 or a = 4
When a = 0, `int_a^(a + 1) x*dx = int_0^1x*dx`
= `[x^2/(2)]_0^1`
= `(1)/(2) - 0`
= `(1)/(2)`
When a = 4, `int_a^(a + 1) d*dx = int_4^5x*dx`
= `[x^2/2]_4^5`
= `(25)/(2) - (16)/(2)`
= `(9)/(2)`.
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