(B) : \(I=\int_{0}^{1} x(1-x)^{5} \cdot d x\) \(=\int(1-x)[1-(1-x)]^{5} \cdot d x\) \({\left[\because \int_{0}^{1} f(x) \cdot d x=\int_{0}^{1} f(a-x) \cdot d x\right]}\) \(=\int_{0}^{1}(1-x) x^{5} \cdot d x=\int_{0}^{1}\left(x^{5}-x^{6}\right) d x\) \(=\left[\frac{x^{6}}{6}\right]_{0}^{1}-\left[\frac{x^{7}}{7}\right]_{0}^{1}\) \(\frac{1}{6}\left(1^{6}-0\right)-\frac{1}{7}\left(1^{7}-0\right)=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)
(B) : \(I=\int_{0}^{1} x(1-x)^{5} \cdot d x\) \(=\int(1-x)[1-(1-x)]^{5} \cdot d x\) \({\left[\because \int_{0}^{1} f(x) \cdot d x=\int_{0}^{1} f(a-x) \cdot d x\right]}\) \(=\int_{0}^{1}(1-x) x^{5} \cdot d x=\int_{0}^{1}\left(x^{5}-x^{6}\right) d x\) \(=\left[\frac{x^{6}}{6}\right]_{0}^{1}-\left[\frac{x^{7}}{7}\right]_{0}^{1}\) \(\frac{1}{6}\left(1^{6}-0\right)-\frac{1}{7}\left(1^{7}-0\right)=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)
(B) : \(I=\int_{0}^{1} x(1-x)^{5} \cdot d x\) \(=\int(1-x)[1-(1-x)]^{5} \cdot d x\) \({\left[\because \int_{0}^{1} f(x) \cdot d x=\int_{0}^{1} f(a-x) \cdot d x\right]}\) \(=\int_{0}^{1}(1-x) x^{5} \cdot d x=\int_{0}^{1}\left(x^{5}-x^{6}\right) d x\) \(=\left[\frac{x^{6}}{6}\right]_{0}^{1}-\left[\frac{x^{7}}{7}\right]_{0}^{1}\) \(\frac{1}{6}\left(1^{6}-0\right)-\frac{1}{7}\left(1^{7}-0\right)=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)
(B) : \(I=\int_{0}^{1} x(1-x)^{5} \cdot d x\) \(=\int(1-x)[1-(1-x)]^{5} \cdot d x\) \({\left[\because \int_{0}^{1} f(x) \cdot d x=\int_{0}^{1} f(a-x) \cdot d x\right]}\) \(=\int_{0}^{1}(1-x) x^{5} \cdot d x=\int_{0}^{1}\left(x^{5}-x^{6}\right) d x\) \(=\left[\frac{x^{6}}{6}\right]_{0}^{1}-\left[\frac{x^{7}}{7}\right]_{0}^{1}\) \(\frac{1}{6}\left(1^{6}-0\right)-\frac{1}{7}\left(1^{7}-0\right)=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)