1. Tardigrade
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  3. Mathematics
  4. Let J =∫ limits0∞ ( ln x /1+ x 3) dx and K =∫ limits0∞ ( x ln x /1+ x 3) dx then

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Q. Let $J =\int\limits_0^{\infty} \frac{\ln x }{1+ x ^3} dx$ and $K =\int\limits_0^{\infty} \frac{ x \ln x }{1+ x ^3} dx$ then

Integrals

Solution:

$J + K =\int\limits_0^{\infty} \frac{( x +1) \ln x }{1+ x ^3} dx =\int\limits_0^{\infty} \frac{\ln x d x }{ x ^2- x +1} dx$
start $x =\frac{1}{ t } \Rightarrow J + K =-( J + K ) \Rightarrow J + K =0$ Ans.
Note: $J=-\frac{2 \pi^2}{27}$ and $K=\frac{2 \pi^2}{27}$ (this note is not for display to students)