Integrate x^x

Asked by Shantanu Kasture | 31st Oct, 2014, 08:37: PM

Expert Answer:

W e space n e e d space t o space e v a l u a t e space I equals integral x to the power of x d x T a k i n g space n a t u r a l space log a r i t h m s space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma ln I equals ln integral x to the power of x d x equals integral ln x to the power of x d x equals integral x ln x d x U sin g 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 ln I equals open square brackets x squared over 2 cross times ln x close square brackets minus 1 half cross times x squared over 2 rightwards double arrow ln I equals open square brackets x squared over 2 cross times ln x close square brackets minus x squared over 4 rightwards double arrow ln I equals x squared over 4 open square brackets 2 ln x minus 1 close square brackets rightwards double arrow ln I equals x squared over 4 open square brackets ln x squared minus ln open parentheses e x p close parentheses close square brackets rightwards double arrow ln I equals x squared over 4 open square brackets ln fraction numerator x squared over denominator e x p end fraction close square brackets rightwards double arrow ln I equals ln open square brackets fraction numerator x squared over denominator e x p end fraction close square brackets to the power of x squared over 4 end exponent rightwards double arrow I equals open square brackets fraction numerator x squared over denominator e x p end fraction close square brackets to the power of x squared over 4 end exponent plus C

Answered by Vimala Ramamurthy | 1st Nov, 2014, 10:55: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.