Question

Let $J=\underset{0}{\overset{\infty }{\int }}\frac{\ln x}{1+x^3}dx$ and $K=\underset{0}{\overset{\infty }{\int }}\frac{x\ln x}{1+x^3}dx$ then

• A. $J+K=0$
• B. $J-K=0$
• C. J + K $<0$
• D. $J+K>0$

Answer 1

Answer: (A)

$J+K=\underset{0}{\overset{\infty }{\int }}\frac{(x+1)\ln x}{1+x^3}dx=\underset{0}{\overset{\infty }{\int }}\frac{\ln xdx}{x^2-x+1}dx$
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)