Question

Evaluate `int_0^1 x(1 - x)^5  "d"x`

Answer 1

Let I = `int_0^1 x(1 - x)^5  "d"x`

= `int_0^1 (1 - x)[1 - (1 - x)]^5  "d"x`   ......`[because int_0^"a" "f"(x)  "d"x = int_0^"a" "f"("a" - x)  "d"x]`

= `int_0^1(1 - x)x^5  "d"x`

= `int_0^1(x^5 - x^6) "d"x`

= `int_0^1 x^5  "d"x - int_0^1 x^6  "d"x`

= `[x^6/6]_0^1 - [x^7/7]_0^1`

= `1/6 (1^6 - 0) - 1/7 (1^7 - 0)`

= `1/6 - 1/7`

∴ I = `1/42`