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$\int (x - 11_{c_{1}} x^{2} + 11_{c_{2}} x^{3} - 11_{c_{3}} x^{4} + ..... - 11_{c_{11}} x^{12}) d x =$

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$\frac{{(1 - x)}^{13}}{13} - \frac{{(1 - x)}^{12}}{12} + c$

$\frac{{(1 - x)}^{12}}{12} - \frac{{(1 - x)}^{13}}{13} + c$

$\frac{{(1 - x)}^{12}}{12} - \frac{{(1 - x)}^{11}}{11} + c$

$\frac{{(1 - x)}^{11}}{11} - \frac{{(1 - x)}^{12}}{12} + c$

answer is B.

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Detailed Solution

given : $\int x {(1 - x)}^{11} d x$
$\begin{array}{l} = \int 1 - (1 - x) (1 - x)^{11} d x \\ = \int (1 - x)^{11} - (1 - x)^{12} d x \\ = - \frac{(1 - x)^{12}}{12} + \frac{(1 - x)^{13}}{13} + c \end{array}$

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