Mathematics

$$\displaystyle \int_{0}^{1}x\left ( 1-x \right )^{4}dx= \frac{1}{C}$$, then $$C=?$$


ANSWER

30


SOLUTION
Let $$\displaystyle I=\int _{ 0 }^{ 1 } x\left( 1-x \right) ^{ 4 }dx$$
Using $$\int _{ a }^{ b }{ f\left( x \right) dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } $$
$$\displaystyle \therefore I=\int _{ 0 }^{ 1 } \left( 1-x \right) x^{ 4 }dx=\int _{ 0 }^{ 1 } \left( x^{ 4 }-x^{ 5 } \right) dx=\frac { 1 }{ 5 } -\frac { 1 }{ 6 } =\frac { 1 }{ 30 } $$
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