Who's up to the challenge?16

Calculus Level 5

$\displaystyle\int _{ 0 }^{ \infty }{ { e }^{ -x }{ 2 }^{ x }\ln x\, dx=\dfrac { A\gamma +\ln(B-\ln(C)) }{ \ln(D)-F } } \\ \\ A,B,C,D,F\quad \in \mathrm{N}$

find $A+B+C+D+F$ .

Inspiration .

this is a part of Who's up to the challenge?

The answer is 7.

1 solution

Rohan Shinde
Dec 10, 2018

Let $I(a)=\int_0^{\infty} e^{-(1-\ln 2)x} x^a dx$

Hence we need $I'(0)$

It is quite known that $J=\int_0^{\infty} e^{-bx} x^a dx= \frac {\Gamma(a+1)}{b^{a+1}}$

On substituting $b=1-\ln 2$ .

We get $I(a)=\frac {\Gamma(a+1)}{(1-\ln 2)^{a+1}}$

Hence $I'(a) =\frac {\Gamma(a+1)}{(1-\ln 2)^{a+1}}\left(\psi(a+1)-\ln(1-\ln 2)\right)$

Substitute $a=0$ to get the answer.

Who's up to the challenge?16

The answer is 7.

1 solution

0 pending reports