please give me the complete solution of 

Asked by Balbir | 19th Oct, 2017, 11:27: AM

Expert Answer:

begin mathsize 12px style G i v e n space I n t e g r a t i o n space I space equals space I subscript 1 plus I subscript 2 space space equals space integral open parentheses e to the power of a log x end exponent space plus e to the power of x log a end exponent space close parentheses d x
w h e r e space I subscript 1 space equals space integral e to the power of a log x end exponent space d x space a n d space I subscript 2 space equals space integral e to the power of x log a end exponent space d x

space I subscript 1 space equals space integral e to the power of a log x end exponent space d x space equals space integral u d v space semicolon space w h e r e space u space equals space e to the power of a log x end exponent space a n d space d v space equals space d x

d u space equals space e to the power of a log x end exponent cross times space a over x d x space a n d space v space equals space x semicolon
integral u d v space equals space left parenthesis u v right parenthesis space minus space integral v d u

H e n c e space I subscript 1 space equals space open parentheses space space space e to the power of a log x end exponent space x close parentheses minus integral space x space cross times space space e to the power of a log x end exponent cross times space a over x d x space equals space open parentheses space e to the power of a log x end exponent space x close parentheses space minus space a integral e to the power of a log x end exponent d x
space space space space space space space space space space space space space space space space space space space space equals space space open parentheses space e to the power of a log x end exponent space x close parentheses space minus space a I subscript 1

h e n c e space I subscript 1 space equals space fraction numerator space open parentheses space e to the power of a log x end exponent space x close parentheses space over denominator open parentheses a plus 1 close parentheses end fraction end style
 
begin mathsize 12px style I subscript 2 space equals space integral e to the power of x log a end exponent d x

L e t space u space equals space x log a comma space d u space equals space log a cross times d x semicolon

I subscript 2 space equals space integral e to the power of u fraction numerator d u over denominator log a end fraction space equals space fraction numerator 1 over denominator log a end fraction e to the power of x log a end exponent

H e n c e space I space equals space I subscript 1 plus I subscript 2 space equals space fraction numerator x e to the power of a log x end exponent over denominator a plus 1 end fraction space plus space fraction numerator e to the power of x log a end exponent over denominator log a end fraction end style

Answered by  | 14th Dec, 2017, 05:10: PM