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प्रश्न
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)]`
उत्तर
`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)]`.
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