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Question
Evaluate `int_0^1 x(1 - x)^5 "d"x`
Solution
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`
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