Discussion
\(\int x(1-x)^{10}dx =\)
Answer: A
The integral is \(\frac{(1 - x)^{12}}{12} - \frac{(1 - x)^{11}}{11} + C\).
Substitute \(u = 1 - x\), so \(x = 1 - u\) and \(dx = -du\):
\[ \begin{aligned} \int x(1 - x)^{10}dx &= -\int (1 - u)u^{10}du \\ &= \int (u^{11} - u^{10})du \\ &= \frac{u^{12}}{12} - \frac{u^{11}}{11} + C \end{aligned} \]
Putting back \(u = 1 - x\) gives the answer. The '+' version comes from losing the minus sign of dx, and the options with \((1 + x)^{11}\) use the wrong substitution.
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