Question

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

Options

• `1/6`

• `1/7`

• `1/21`

• `1/42`

Answer 1

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

Explanation:'

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

Put 1 - x = t ⇒ dx = - dt

∴ I = `- int_1^0 (1 - "t")"t"^5 "dt" = int_0^1 ("t"^5 - "t"^6) "dt"   ...[because int_"a"^"b" "f"(x) "dx" = - int_"b"^"a" "f"(x) "dx"]`

`= ["t"^6/6 - "t"^7/7]_0^1 = 1/6 - 1/7`

∴ I = `1/42`