- #1
DaalChawal
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Integration Doubt: Answers & Solutions
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In summary, the conversation was about simplifying the integrand, which involved using substitution and differentiation to reach a simpler form. The final form was a fraction with a numerator and denominator that both contain terms involving natural logarithms and exponentials.
- #2
Prove It
Gold Member
MHB
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It might be worth simplifying the integrand...
$\displaystyle \begin{align*} \frac{\ln{\left( \mathrm{e}\,x^{x+1} \right)} + \left[ \ln{ \left( x^{\sqrt{x}} \right) } \right]^2 }{1 + x\ln{ \left( x \right) } \ln{ \left( \mathrm{e}^x\,x^x \right) }} &= \frac{ \ln{\left( \mathrm{e} \right) } + \ln{ \left( x^{x+1} \right) } + \left[ \sqrt{x} \, \ln{ \left( x \right) } \right] ^2 }{ 1 + x\ln{ \left( x \right) } \left[ \ln{\left( \mathrm{e}^x \right) } + \ln{ \left( x^x \right) } \right] } \\ &= \frac{ 1 + \left( x + 1 \right) \ln{ \left( x \right) } + x \, \left[ \ln{ \left( x \right) } \right] ^2 }{ 1 + x \ln{ \left( x \right) } \left[ x + x\ln{ \left( x \right) } \right] } \end{align*}$
I don't know if this helps, but it looks simpler at least...
$\displaystyle \begin{align*} \frac{\ln{\left( \mathrm{e}\,x^{x+1} \right)} + \left[ \ln{ \left( x^{\sqrt{x}} \right) } \right]^2 }{1 + x\ln{ \left( x \right) } \ln{ \left( \mathrm{e}^x\,x^x \right) }} &= \frac{ \ln{\left( \mathrm{e} \right) } + \ln{ \left( x^{x+1} \right) } + \left[ \sqrt{x} \, \ln{ \left( x \right) } \right] ^2 }{ 1 + x\ln{ \left( x \right) } \left[ \ln{\left( \mathrm{e}^x \right) } + \ln{ \left( x^x \right) } \right] } \\ &= \frac{ 1 + \left( x + 1 \right) \ln{ \left( x \right) } + x \, \left[ \ln{ \left( x \right) } \right] ^2 }{ 1 + x \ln{ \left( x \right) } \left[ x + x\ln{ \left( x \right) } \right] } \end{align*}$
I don't know if this helps, but it looks simpler at least...
- #3
DaalChawal
- 87
- 0
I also reached this step I tried to create differentiation inside and used substitution too but could not solve it further.
FAQ: Integration Doubt: Answers & Solutions
What is integration doubt?
Integration doubt refers to the uncertainty or confusion that arises when solving integration problems in mathematics. It is common for students to struggle with integration, as it involves finding the antiderivative or integral of a function.
How can I overcome integration doubt?
One of the best ways to overcome integration doubt is through practice. The more you practice solving integration problems, the more comfortable and confident you will become. It is also helpful to review and understand the basic principles and rules of integration.
What are some common mistakes to avoid when solving integration problems?
Some common mistakes to avoid when solving integration problems include forgetting to add the constant of integration, incorrect use of substitution or integration by parts, and not simplifying the final answer. It is important to double-check your work and be mindful of these mistakes.
Are there any online resources that can help with integration doubt?
Yes, there are many online resources available to help with integration doubt. Some popular options include Khan Academy, Wolfram Alpha, and online forums and communities where you can ask for help and clarification on specific integration problems.
How can I improve my understanding of integration concepts?
Improving your understanding of integration concepts involves actively engaging with the material. This can include attending lectures, taking notes, practicing problems, and seeking help when needed. It can also be helpful to connect integration to real-world applications to better understand its significance and relevance.