Question

The integral $integral fraction numerator dx over denominator straight x squared space left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction space equals$

$open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C$
$left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C$
$negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C$
$negative open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C$

Solution

negative left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of 1 fourth end exponent space plus straight C

negative open parentheses fraction numerator straight x to the power of 4 plus 1 over denominator straight x to the power of 4 end fraction close parentheses to the power of 1 fourth end exponent space plus straight C

integral fraction numerator dx over denominator straight x squared left parenthesis straight x to the power of 4 plus 1 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction space equals space integral fraction numerator dx over denominator straight x to the power of 5 open parentheses 1 plus begin display style 1 over straight x to the power of 4 end style close parentheses to the power of begin display style 3 over 4 end style end exponent end fraction Put space space 1 space plus space 1 over straight x to the power of 4 equals space straight t to the power of 4 minus 4 over straight x to the power of 5 dx space equals space 4 straight t cubed dt rightwards double arrow space dx over straight x to the power of 5 space equals space minus space straight t cubed dt space equals space integral fraction numerator negative straight t cubed dt over denominator straight t cubed end fraction equals space minus integral dt space equals space minus space minus straight t plus straight c space equals space minus space open parentheses 1 plus 1 fourth close parentheses to the power of 1 fourth end exponent space plus space straight C

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