Integrating e^x(x+1)lnx Using Integration by Parts

In summary, integration by parts is a method used in calculus to evaluate the integral of a product of two functions by choosing one function as "u" and the other as "dv". It is typically used when the integrand is difficult to integrate using other techniques. The general rule for choosing "u" and "dv" is "LIATE" and the steps for solving an integration by parts problem include identifying "u" and "dv", finding "du" and "v", and using the integration by parts formula. To avoid mistakes, it is important to choose "u" and "dv" carefully, double check the steps, use parentheses, and check the final answer with differentiation.
  • #1
BrownianMan
134
0
[tex]\[\int e^x(x+1)\ln x \ dx \][/tex]

Not sure how to approach this. Would I have to multiply it out first?
 
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  • #2
BrownianMan said:
[tex]\[\int e^x(x+1)\ln x \ dx \][/tex]

Not sure how to approach this. Would I have to multiply it out first?
For starters, I would split it into two integrals and see if integration by parts works on each one.

[tex]\[\int e^x(x+1)\ln x dx = \int xe^x~lnx~dx + \int e^x~lnx~dx[/tex]
 
  • #3
Thanks! I got it.
 

Related to Integrating e^x(x+1)lnx Using Integration by Parts

1. What is integration by parts?

Integration by parts is a method used in calculus to evaluate the integral of a product of two functions. It is based on the product rule of differentiation, and involves choosing one function to be the "u" in the rule and the other function to be the "dv".

2. When should I use integration by parts?

Integration by parts is typically used when the integrand (the function being integrated) is a product of two functions, and it is difficult to find an antiderivative using other integration techniques such as substitution or trigonometric identities.

3. How do I choose which function to be the "u" and which to be the "dv"?

The general rule is to choose "u" to be the function that becomes simpler when differentiated, and "dv" to be the function that becomes easier to integrate. A common acronym used to remember this is "LIATE" which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions.

4. What are the steps for solving an integration by parts problem?

The steps for solving an integration by parts problem are:

  1. Identify "u" and "dv" in the integrand
  2. Find "du" by differentiating "u"
  3. Find "v" by integrating "dv"
  4. Substitute all values into the integration by parts formula: ∫udv = uv - ∫vdu
  5. Solve for the final integral ∫vdu

5. Are there any tips for avoiding mistakes when using integration by parts?

Yes, there are a few tips that can help avoid mistakes when using integration by parts:

  • Choose "u" and "dv" carefully
  • Double check the differentiation and integration steps
  • Use parentheses to avoid sign errors
  • Check your final answer by differentiating it

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