# ∫xx(1+logx)dx

Video Solution
Text Solution
Generated By DoubtnutGPT

## To solve the integral ∫xx(1+logx)dx, we can follow these steps:Step 1: SubstitutionLet t=xx. To differentiate this, we can use the property of logarithms:logt=xlogxNow, differentiate both sides with respect to x:ddx(logt)=1tdtdx=logx+1Thus, we have:dtdx=t(logx+1)Step 2: Express dx in terms of dtFrom the equation above, we can express dx as:dx=dtt(logx+1)Step 3: Substitute back into the integralNow, substitute t=xx and dx into the integral:∫xx(1+logx)dx=∫t⋅dtt(logx+1)This simplifies to:∫dt=t+CStep 4: Substitute back for tNow, substitute back t=xx:∫xx(1+logx)dx=xx+CFinal AnswerThus, the final answer is:∫xx(1+logx)dx=xx+C---

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Updated on:7/8/2024

### Knowledge Check

• Question 1 - Select One

## Assertion (A) :∫exx(1+xlogx)dx=exlogx+c. Reason (R) :∫ex[f(x)+f'(x)]dx=exf(x)+c

ABoth A and R are true and R is the correct explanation of A
BBoth A and R are true but R is not the correct explanation of A
CA is true but R is false
DA is false but R is true

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NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation
The text passage contains a mathematical problem involving an integral and its solution, followed by a multiple-choice question related to a similar integral problem.
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
# ∫xx(1+logx)dx

Video Solution
Text Solution
Generated By DoubtnutGPT

## To solve the integral ∫xx(1+logx)dx, we can follow these steps:Step 1: SubstitutionLet t=xx. To differentiate this, we can use the property of logarithms:logt=xlogxNow, differentiate both sides with respect to x:ddx(logt)=1tdtdx=logx+1Thus, we have:dtdx=t(logx+1)Step 2: Express dx in terms of dtFrom the equation above, we can express dx as:dx=dtt(logx+1)Step 3: Substitute back into the integralNow, substitute t=xx and dx into the integral:∫xx(1+logx)dx=∫t⋅dtt(logx+1)This simplifies to:∫dt=t+CStep 4: Substitute back for tNow, substitute back t=xx:∫xx(1+logx)dx=xx+CFinal AnswerThus, the final answer is:∫xx(1+logx)dx=xx+C---

|
Updated on:7/8/2024

### Knowledge Check

• Question 1 - Select One

## Assertion (A) :∫exx(1+xlogx)dx=exlogx+c. Reason (R) :∫ex[f(x)+f'(x)]dx=exf(x)+c

ABoth A and R are true and R is the correct explanation of A
BBoth A and R are true but R is not the correct explanation of A
CA is true but R is false
DA is false but R is true

Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc
NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation
user
The text passage contains a mathematical problem involving an integral and its solution, followed by a multiple-choice question related to a similar integral problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER