Ex 7.11, 7 - Evaluate using properties x (1 - x)n dx

Ex 7.11,7 By using the properties of definite integrals, evaluate the integrals : 0﷮1﷮ 𝑥 1−𝑥﷯﷮𝑛﷯﷯𝑑𝑥 Let I= 0﷮1﷮𝑥 1−𝑥﷯﷮𝑛﷯ 𝑑𝑥﷯ ∴ I= 0﷮1﷮ 1−𝑥﷯ 1− 1−𝑥﷯﷯﷮𝑛﷯ 𝑑𝑥﷯ I= 0﷮1﷮ 1−𝑥﷯ 1−1+𝑥﷯﷮𝑛﷯ 𝑑𝑥﷯ I= 0﷮1﷮ 1−𝑥﷯ 𝑥﷯﷮𝑛﷯ 𝑑𝑥﷯ I= 0﷮1﷮ 1−𝑥﷯ 𝑥﷮𝑛﷯ 𝑑𝑥﷯ I= 0﷮1﷮ 𝑥﷮𝑛﷯− 𝑥﷮𝑛 + 1﷯﷯ 𝑑𝑥﷯ I= 0﷮1﷮ 𝑥﷮𝑛﷯ 𝑑𝑥﷯− 0﷮1﷮ 𝑥﷮𝑛 + 1﷯ 𝑑𝑥﷯ I= 𝑥﷮𝑛 + 1﷯﷮𝑛 + 1﷯﷯﷮0﷮1﷯− 𝑥﷮𝑛 + 2﷯﷮𝑛 + 2﷯﷯﷮0﷮1﷯ I= 1﷯﷮𝑛 + 1﷯﷮𝑛 + 1﷯− 0﷯﷮𝑛 + 1﷯﷮𝑛 + 1﷯− 1﷯﷮𝑛 + 2﷯﷮𝑛 + 2﷯− 0﷯﷮𝑛 + 2﷯﷮𝑛 + 2﷯﷯ I= 1﷮𝑛 + 1﷯− 1﷮𝑛 + 2﷯ I= 𝑛 + 2 − 𝑛 + 1﷯﷮ 𝑛 + 1﷯ 𝑛 + 2﷯﷯ I= 1﷮ 𝑛 + 1﷯ 𝑛 + 2﷯﷯ ∴ 0﷮1﷮𝑥 (1−𝑥)﷮𝑛﷯𝑑𝑥﷯ = 𝟏﷮ 𝒏 + 𝟏﷯ 𝒏 + 𝟐﷯﷯