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`

Answer 1

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)`.