can u please only tell the answer of integration of dx/(xlogx)2 \
i know ans is in gamma function but sir pls tell me only solution

Asked by Devesh | 22nd Sep, 2015, 06:04: PM

Expert Answer:

I equals integral fraction numerator d x over denominator open parentheses x log open parentheses x close parentheses close parentheses squared end fraction F i r s t space c h a n g e space t h e space b a s e space t o space n a t u r a l space l o a g r i t h m. T h u s comma space I equals open parentheses ln open parentheses 10 close parentheses close parentheses squared cross times integral fraction numerator d x over denominator x squared ln squared open parentheses x close parentheses end fraction A p p l y i n g space i n t e g r a l space s u b s t i t u t i i o n space t equals ln open parentheses x close parentheses comma space t h e n space d t equals open parentheses ln open parentheses x close parentheses close parentheses apostrophe space d x space a n d space fraction numerator d x over denominator x end fraction equals d t T h u s comma space I equals open parentheses l n open parentheses 10 close parentheses close parentheses squared integral fraction numerator 1 over denominator t squared e to the power of t end fraction d t space space space space space space space space space space space equals open parentheses l n open parentheses 10 close parentheses close parentheses squared integral e to the power of negative t end exponent over t squared d t a n d space i n t e g r a t i o n space b y space p a r t s comma space w e space h a v e comma space u equals e to the power of negative t end exponent semicolon space u apostrophe equals negative e to the power of negative t end exponent semicolon v apostrophe equals 1 over t squared semicolon v equals negative 1 over t T h u s comma space I equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets e to the power of negative t end exponent open parentheses negative 1 over t close parentheses minus integral open parentheses negative 1 over t close parentheses open parentheses negative e to the power of negative t end exponent close parentheses d t close square brackets space space space space space space space space space space space equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets open parentheses negative e to the power of negative t end exponent over t close parentheses minus integral open parentheses e to the power of negative t end exponent over t close parentheses d t close square brackets space space space space space space space space space space space equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets open parentheses negative e to the power of negative t end exponent over t close parentheses minus integral open parentheses e to the power of negative t end exponent over t close parentheses d t close square brackets A g a i n space s u b s t i t u t i n g space s equals negative t semicolon space d s equals negative 1 cross times d t rightwards double arrow d t equals negative 1 cross times d s T h u s comma space I equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets open parentheses negative e to the power of negative t end exponent over t close parentheses minus open parentheses negative integral open parentheses fraction numerator e to the power of s over denominator negative s end fraction close parentheses d s close parentheses close square brackets space space space space space space space space space space space space equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets open parentheses negative e to the power of negative t end exponent over t close parentheses minus open parentheses integral open parentheses e to the power of s over s close parentheses d s close parentheses close square brackets C o n s i d e r space integral open parentheses e to the power of s over s close parentheses d s space equals E i open parentheses s close parentheses equals E i open parentheses negative t close parentheses equals E i open parentheses negative ln left parenthesis x right parenthesis close parentheses T h u s comma space I equals open parentheses l n open parentheses 10 close parentheses close parentheses squared open square brackets open parentheses negative e to the power of negative t end exponent over t close parentheses minus E i open parentheses negative l n left parenthesis x right parenthesis close parentheses close square brackets plus C G a m m a space f u n c t i o n space i s space capital gamma open parentheses t close parentheses equals integral subscript 0 superscript infinity x to the power of t minus 1 end exponent e to the power of negative x end exponent d x

Answered by Vimala Ramamurthy | 23rd Sep, 2015, 11:59: AM