Question

\[\int\limits_0^1 x \left( 1 - x \right)^5 dx\]

Answer 1

\[Let\ I = \int_0^1 x \left( 1 - x \right)^5 d x . Then, \]
\[I = \int_0^1 \left( x - 1 + 1 \right) \left( 1 - x \right)^5 d x\]
\[ \Rightarrow I = \int_0^1 \left[ - \left( 1 - x \right)^6 + \left( 1 - x \right)^5 \right] d x\]
\[ \Rightarrow I = \left[ \frac{\left( 1 - x \right)^7}{7} \right]_0^1 - \left[ \frac{\left( 1 - x \right)^6}{6} \right]_0^1 \]
\[ \Rightarrow I = - \frac{1}{7} + \frac{1}{6}\]
\[ \Rightarrow I = \frac{1}{42}\]