207 views

$$\int_{0}^{1} \ln x\, \mathrm{d}x=$$

1. $1$
2. $-1$
3. $\infty$
4. $-\infty$
5. None of the above.
asked in Calculus | 207 views

Use Integration by Parts

ln(x) dx

set
u = ln(x),    dv = dx
then we find
du = (1/x) dx,    v = x

substitute

ln(x) dx =  u dv

and use integration by parts

= uv -  v du

substitute u=ln(x), v=x, and du=(1/x)dx

= ln(x) x -  x (1/x) dx
= ln(x) x -  dx
= ln(x) x - x + C
= x ln(x) - x + C.

Now Put Limits

[ln(1)-1+C]-[0-0+C]= -1

Note-Lim [xlnx] = 0.
x->0

selected by
ans -1

but note 1 thing it is definite integral not contains constatnt
ln0 is -infinity then 0*-infinity =0 ??
The text passage contains a mathematical question in the form of a definite integral and provides a detailed solution, including the final answer.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
207 views

$$\int_{0}^{1} \ln x\, \mathrm{d}x=$$

1. $1$
2. $-1$
3. $\infty$
4. $-\infty$
5. None of the above.
asked in Calculus | 207 views

Use Integration by Parts

ln(x) dx

set
u = ln(x),    dv = dx
then we find
du = (1/x) dx,    v = x

substitute

ln(x) dx =  u dv

and use integration by parts

= uv -  v du

substitute u=ln(x), v=x, and du=(1/x)dx

= ln(x) x -  x (1/x) dx
= ln(x) x -  dx
= ln(x) x - x + C
= x ln(x) - x + C.

Now Put Limits

[ln(1)-1+C]-[0-0+C]= -1

Note-Lim [xlnx] = 0.
x->0

selected by
ans -1

but note 1 thing it is definite integral not contains constatnt
ln0 is -infinity then 0*-infinity =0 ??
user
The text passage contains a mathematical question in the form of a definite integral and provides a detailed solution, including the final answer.

TYPE: SINGLE_QUESTION_WITH_ANSWER