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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. Integral 0 to 1 x(1 - x)^n dx

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

    Q7.    \int^1_0x(1-x)^ndx

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We have                 I\ =\ \int^1_0x(1-x)^ndx

Using the property : -

                                        \ \int_0^a\ f(x) dx\ =\ \ \int_0^a\ f(a-x) dx

We get : - 

                                 I\ =\ \int^1_0x(1-x)^ndx\ =\ \int^1_0(1-x)(1-(1-x))^ndx

or                              I\ =\ \int^1_0(1-x)x^n\ dx

or                             I\ =\ \int^1_0(x^n\ -\ x^{n+1}) \ dx

or                                    =\ \left [ \frac{x^{n+1}}{n+1}\ -\ \frac{x^{n+2}}{n+2} \right ]^1_0

or                                     =\ \left [ \frac{1}{n+1}\ -\ \frac{1}{n+2} \right ]

or                             I\ =\ \frac{1}{(n+1)(n+2)}

Posted by

Devendra Khairwa

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