For x > 0 , let f x = ∫ 1 x log ⁡ t 1 + t   d t .   Then f x + f 1 x is equal to

Question

For x &#62; 0 , let f x = &#8747; 1 x log &#8289; t 1 + t &#160; d t . &#160; Then f x + f 1 x is equal to

Aicharya · Accepted Answer

f x = &#160; &#8747; 1 x log &#8289; t 1 + t &#160; d t &#8658; f x = &#8747; 1 x log &#8289; z ( 1 + z ) d z And f 1 x = &#160; &#8747; 1 1 x log &#8289; t 1 + t &#160; d t Put t = 1 z d t = &#160; - 1 z 2 &#160; d z f x = &#160; &#8747; 1 x log &#8289; z z 2 1 + &#160; 1 z &#160; &#8901; d z f x = &#160; &#8747; 1 x log &#8289; z z ( 1 + z ) &#160; &#160; d z f x + f 1 x = &#160; &#8747; 1 x log &#8289; z &#160; 1 1 + z + 1 z ( 1 + z ) &#160; d z = &#160; &#8747; 1 x 1 z log &#8289; z &#160; d z Put log &#8289; z = P &#8658; 1 z &#160; d z = d P &#8747; 1 x P &#8901; d P = P 2 2 1 x = 1 2 &#160; log 2 &#8289; z 1 x = &#160; log &#8289; x 2 2

For x &#62; 0 , let f x = &#8747; 1 x log &#8289; t 1 + t &#160; d t . &#160; Then f x + f 1 x is equal to

Solution

For x > 0 , let f x = ∫ 1 x log ⁡ t 1 + t d t . Then f x + f 1 x is equal to