text: # ∫xx(1+logx)dx
  
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  ## 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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