Integral of e^(x ln(x)) (1 + ln(x)) from 0 to 1

[rare shoal]

I need some guidance on how to solve this type of integrals.
I found that i can simplify it into: x^x (1 + ln(x)) and then study the 2 integrals:
-Integral x^x from 0 to 1
-Integral x^x*ln(x) from 0 to 1
Now i'm stack and unsure how to proceed with the evaluation of these integrals analytically.
Is there an analytical method or a standard result that I can use to solve these integrals, or should I consider using numerical approximation techniques? If so, what is the most efficient way to handle these integrals in an exam setting ?

[keen vale]

Do a u-sub

u = x ln x

[rare shoal]

is correct that i have du = ln(x)+1 ?
if so my integral becomes: e^u du and my final resul is e-1 ?

[keen vale]

Lemme do the integral rq and see

[keen vale]

I get that the integral is 0

The u-sub maps the interval over which we integrate to a single point

The mistake you made is likely that you forgot to change your bounds when doing the substitution

You need to take the limit of u as x goes to each bound

[rare shoal]

yep i still integrated from 0 to 1

0 seems the correct answer

[keen vale]

Real quick, would u know how to change the bounds for x ln x as x goes to 0?

$\lim_{x \to 0} x\ln{x}$ isn't immediately apparent

[modest shuttle]

hiidostuff

[rare shoal]

i think i should use De l'Hopital

thank you, you opened my eyes (and saved me)

[keen vale]

no problem

if all else fails a clever u-sub will work