Question

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 1 space straight x space left parenthesis 1 minus straight x right parenthesis to the power of straight n space dx$

Solution

Let I = $integral subscript 0 superscript 1 straight x left parenthesis 1 minus straight x right parenthesis to the power of straight n space dx space equals space integral subscript 0 superscript 1 left parenthesis 1 minus straight x right parenthesis space open curly brackets 1 minus left parenthesis 1 minus straight x right parenthesis close curly brackets to the power of straight n space dx$
$open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$equals space integral subscript 0 superscript 1 left parenthesis 1 minus straight x right parenthesis. space space straight x to the power of straight n space dx space equals space integral subscript 0 superscript 1 left parenthesis straight x to the power of straight n minus straight x to the power of straight n plus 1 end exponent right parenthesis space dx space equals space open square brackets fraction numerator straight x to the power of straight n plus 1 end exponent over denominator straight n plus 1 end fraction minus fraction numerator straight x to the power of straight n plus 2 end exponent over denominator straight n plus 2 end fraction close square brackets subscript 0 superscript 1 equals space open parentheses fraction numerator 1 over denominator straight n plus 1 end fraction minus fraction numerator 1 over denominator straight n plus 2 end fraction close parentheses space minus space left parenthesis 0 minus 0 right parenthesis space equals space fraction numerator 1 over denominator straight n plus 1 end fraction minus fraction numerator 1 over denominator straight n plus 2 end fraction equals space fraction numerator straight n plus 2 minus straight n minus 1 over denominator left parenthesis straight n plus 1 right parenthesis thin space left parenthesis straight n plus 2 right parenthesis end fraction space equals space fraction numerator 1 over denominator left parenthesis straight n plus 1 right parenthesis thin space left parenthesis straight n plus 2 right parenthesis end fraction$

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Integrals

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 1 space straight x space left parenthesis 1 minus straight x right parenthesis to the power of straight n space dx$

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