Evaluate the following definite Integral
∫x * ln(x) dx from 1 to e^(2)
I used the DI method which gave me
D I
+ ln(x) x
- 1/x (x^2)/2
= (ln(x)*x^(2))/2 - ∫ 1/x * x^(2)/2
= (ln(x)*x^(2))/2 - ∫ x/2
= (ln(x)*x^(2))/2 - x^(2)/4 + C
(ln(x)*x^(2))/2 - x^(2)/4| from 1 to e^(2)
= (ln(e^(2))*e^(2)^(2))/2 - e^(2)^(2)/4 - [ (ln(1)*1^(2))/2 - 1^(2)/4 ]
= (ln(e^(2))*e^(4)/2 - e^(4)/4 - [ (0*1)/2 - 1/4 ]
= (1*2)*e^(4)/2 - e^(4)/4 - [ 0 - 1/4 ]
= 2*e^(4)/2 - e^(4)/4+ 1/4
The teacher answer is
e^(2)/2 - e^(4)/4 + 1/4
Where did I go wrong? Thank you!
= 2*e^(4)/2 - e^(4)/4+ 1/4
= 2e^(4)/2 - e^(4)/4 + 1/4
= 2*2e^(4)/4 - e^(4)/4 + 1/4
= 4e^(4)/4 - e^(4)/4 + 1/4
= 3e^(4)/4 + 1/4
= 1/4 [3e^(4) +1]
Alright, thank you. I just couldnt see it somehow
But wait... is the teacher answer: e^(2)/2 - e^(4)/4 + 1/4, a mistake? like the e^(2)/2 part?
It seems that you made a mistake when evaluating the definite integral at the upper limit, e^2. Let's go through the calculations step by step to identify the error.
You correctly used the DI method and obtained the antiderivative:
∫x * ln(x) dx = (ln(x) * x^2)/2 - x^2/4 + C
Next, we substitute the limits:
[(ln(x) * x^2)/2 - x^2/4] | from 1 to e^2
Let's evaluate it at the upper limit first, e^2:
((ln(e^2) * e^2^2)/2 - e^2^2/4)
Now, the issue arises when evaluating ln(e^2). The natural logarithm function ln(x) is the inverse of the exponential function e^x. Therefore, ln(e^2) should simply equal to 2, not 1.
Corrected calculation:
((ln(e^2) * e^2^2)/2 - e^2^2/4)
= (2 * e^4/2 - e^4/4)
= e^4 - e^4/4
= 4e^4/4 - e^4/4
= 3e^4/4
So, the correct answer is 3e^4/4, which matches the teacher's answer: e^2/2 - e^4/4 + 1/4.
∫x * ln(x)
use integration by parts. That is just the product rule for derivatives, only in reverse. So, let
u = lnx
du = 1/x dx
dv = x dx
v = 1/2 x^2
∫u dv = uv - ∫v du
∫x lnx dx = 1/2 x^2 lnx - ∫1/2 x dx
= 1/2 x^2 lnx - 1/4 x^2 = 1/4 x^2 (2lnx - 1)
∫[1,e^2] x lnx dx = (1/4 e^4 (2lne^2 - 1) - (1/4 (2ln1 -1))
= 1/4 e^4 (3) - 1/4 (-1)
= 1/4 (3e^4 + 1)