Question

`int_0^1 (1 - x)^5`dx = ______.

विकल्प

• `1/5`

• `1/42`

• `1/13`

• `13/42`

Answer 1

`int_0^1 (1 - x)^5`dx = `underline(1/42)`.

Explanation:

Let I = `int_0^1 x(1 - x)^5`dx

put 1 - x = t ⇒ x = 1 - t

⇒ - dx = dt

⇒ dx = - dt

when x = 0, then t = 1 and

when x = 1 , then t = 0

`therefore "I" = int_1^0 (1 - "t")("t")^5(- "dt")`

`= int_0^1 ("t"^5 - "t"^6) "dt"`

`= ("t"^6/6 - "t"^7/7)_0^1`

`= 1/6 (1)^6 - 1/7(1)^7`

`= (7 - 6)/42`

`= 1/42`