Question

Evaluate the following : If f(x) = a + bx + cx², show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`

Answer 1

`int_0^1 f(x)*dx = int_0^1 (a + bx + cx^2)*dx`

= `a int_0^1 1*dx + b int_0^1 x*dx + c int_0^1 x^2*dx`

= `[ax + "bx"^2/2 + "cx"^3/3]_0^1`

= `a + b/2 + c/3`                                 ...(1)

Now, `f(0) = a + b(0) + c(0)^2` = a

`f(1/2) = a + b(1/2) + c(1/2)^2 = a + b/2 + c/4`
and 
`f(1) = a + b(1) + c(1)^2` = a + b + c

∴ `(1)/(6)[f(0) + 4f(1/2) + f(1)]`

= `(1)/(6)[a + 4(a + b/2 + c/4) + (a + b + c)]`

= `(1)/(6)[a + 4a + 2b + c + a + b + c]`

= `(1)/(6)[6a + 3b + 2c]`

= `a + b/2 + c/3`                               ...(2)
∴ from (1) and (2),

`int_0^1 f(x)*dx = (1)/(6)[f(0) + 4f(1/2) + f(1)]`.