Question

The value of the integral $integral subscript 1 third end subscript superscript 1 fraction numerator left parenthesis straight x minus straight x cubed right parenthesis to the power of begin display style 1 third end style end exponent over denominator straight x to the power of 4 end fraction dx$

6

0

3

4

Solution

Let $straight I space equals space integral subscript 1 third end subscript superscript 1 space fraction numerator left parenthesis straight x minus straight x cubed right parenthesis to the power of begin display style 1 third end style end exponent over denominator straight x to the power of 4 end fraction dx space equals space integral subscript 3 superscript 1 fraction numerator straight x open parentheses begin display style 1 over straight x squared end style minus 1 close parentheses to the power of begin display style 1 third end style end exponent over denominator straight x to the power of 4 end fraction dx space space space$
$equals space integral subscript 1 third end subscript superscript 1 open parentheses begin display style 1 over straight x squared end style minus 1 close parentheses to the power of begin display style 1 third end style end exponent over straight x cubed dx space equals space integral subscript 1 third end subscript superscript 1 space open parentheses 1 over straight x squared minus 1 close parentheses to the power of 1 third end exponent. space 1 over straight x cubed dx$
Put $1 over straight x squared space equals space straight t comma space space space space space therefore space space space minus 2 over straight x cubed dx space equals space dt space space space rightwards double arrow space space space 1 over straight x cubed dx space equals space minus 1 half dt$
When $straight x space equals space 1 third comma space space straight t space equals space 9$
When x = 1, t = 1
$therefore$ $straight I space equals negative 1 half integral subscript 9 superscript 1 left parenthesis straight t minus 1 right parenthesis to the power of 1 third end exponent dt space equals space minus 1 half open square brackets fraction numerator left parenthesis straight t minus 1 right parenthesis to the power of begin display style 4 over 3 end style end exponent over denominator begin display style 4 over 3 end style end fraction close square brackets subscript 0 superscript 1$
$equals negative 3 over 8 open square brackets left parenthesis straight t minus 1 right parenthesis to the power of 4 over 3 end exponent close square brackets subscript 9 superscript 1 space equals space minus 3 over 8 open square brackets left parenthesis 1 minus 1 right parenthesis to the power of 4 over 3 end exponent minus left parenthesis 9 minus 1 right parenthesis to the power of 4 over 3 end exponent close square brackets space equals space minus 3 over 8 open square brackets 0 minus left parenthesis 8 right parenthesis to the power of 4 over 3 end exponent close square brackets space equals space 3 over 8 left parenthesis 2 cubed right parenthesis to the power of 4 over 3 end exponent space equals space 3 over 8 cross times 2 to the power of 4 space equals space 3 over 8 cross times 16 space equals space 6$

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