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Question

$$\int _{ 0 }^{ 1 }{ x{ \left[ 1-x \right]  }^{ 11 } } dx=........$$


A
1132
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B
1156
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C
1121
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D
112
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Solution

The correct option is A $$-\cfrac { 1 }{ 156 } $$
$$\rightarrow I=\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 11 } } dx$$
Take $$1-x$$ is place of $$x$$
$$=\int _{ 0 }^{ 1 }{ (1-x){ \left[ 1-(1-x) \right]  }^{ 11 }dx } $$
$$=-\int _{ 0 }^{ 1 }{ \left( 1-x \right) { x }^{ 11 } } dx=-\int _{ 0 }^{ 1 }{ \left( { x }^{ 11 }-{ x }^{ 12 } \right)  } dx=-{ \left[ \cfrac { { x }^{ 12 } }{ 12 } -\cfrac { { x }^{ 13 } }{ 13 }  \right]  }_{ 0 }^{ 1 }=\cfrac { 1 }{ 13 } -\cfrac { 1 }{ 12 } $$
$$\therefore I=-\cfrac { 1 }{ 156 } $$

Mathematics

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