Evaluate the following integral:
$integral subscript 4 superscript 9 fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx$

Solution

Let I = $integral subscript 4 superscript 9 fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx$
Let $straight I subscript 1 space equals space integral fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx$
Put    $30 minus straight x to the power of 3 over 2 end exponent space equals space straight t comma space space space therefore space space minus 3 over 2 straight x to the power of 1 half end exponent ax space equals space dt space space space space space space space rightwards double arrow space space space space square root of straight x space dx space space equals negative 2 over 3 dt$
$therefore$    $straight I subscript 1 space equals space minus 2 over 3 integral 1 over straight t squared dt space equals negative 2 over 3 integral straight t to the power of negative 2 end exponent dt space equals space minus 2 over 3 fraction numerator straight t to the power of negative 1 end exponent over denominator negative 1 end fraction space equals space fraction numerator 2 over denominator 3 straight t end fraction$
$therefore$    $integral fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx space equals space fraction numerator 2 over denominator 3 open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses end fraction space equals space straight F left parenthesis straight x right parenthesis comma space say.$
by the second fundamental theorem,
I = F(9) - F(4) = $2 over 3 open square brackets fraction numerator 1 over denominator 30 minus 9 to the power of begin display style 3 over 2 end style end exponent end fraction close square brackets minus 1 third open square brackets fraction numerator 1 over denominator 30 minus 4 to the power of begin display style 3 over 2 end style end exponent end fraction close square brackets$
$equals space 2 over 3 open square brackets fraction numerator 1 over denominator 30 minus 27 end fraction close square brackets minus 2 over 3 open square brackets fraction numerator 1 over denominator 30 minus 8 end fraction close square brackets space equals 2 over 3 cross times 1 third minus 2 over 3 cross times 1 over 22 equals space 2 over 3 open parentheses 1 third minus 1 over 22 close parentheses space equals space 2 over 3 cross times fraction numerator 22 minus 3 over denominator 66 end fraction space equals space 2 over 3 cross times 19 over 66 space equals space 19 over 99$

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Integrals

Evaluate the following integral:
$integral subscript 4 superscript 9 fraction numerator square root of straight x over denominator open parentheses 30 minus straight x to the power of begin display style 3 over 2 end style end exponent close parentheses squared end fraction dx$

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