Question

दर्शाइए कि `int_0^a f(x) g(x) dx = 2 int_0^a f(x)` dx, यदि f और g को f(x) = f(a - x) एवं g(x) + g(a - x) = 4 के रूप में परिभाषित किया गया है।

Answer 1

माना `I = int_0^a f(x) g(x) dx`

`= int_0^a f(a - x) [4 - g(a - x)] dx`

`= 4 int_0^a f(a - x) dx - int_0^a f(a - x) g (a - x)  dx`

माना a - x = t

⇒ - dx = dt

जब x = 0, t = a

तथा x = a, t = 0

`I = -4 int_a^0 f (t) dt + int_a^0 f (t) g (t) dt`

`= 4 int_0^a f (t) dt - int_0^a f (t) g (t)  dt`

`= 4 int_0^a f (x) dx - int_0^a f (x)g (x) dx `

`= 4 int_0^a f (x) dx - I`

⇒ `2I = 4 int_0^a f (x) dx`

अत: `I = 2 int_0^a f (x) dx`