U-substitution and back substitution | Summary and Q&A

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December 28, 2012
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Khan Academy
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U-substitution and back substitution

TL;DR

By making a substitution and simplifying the expression, we can easily integrate complex polynomials.

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Key Insights

  • 💄 Making a substitution in integration can simplify complex expressions, making them easier to integrate.
  • 😑 The choice of substitution may require some trial and error, with a focus on finding a form that simplifies the expression the most.
  • 🥺 Substituting u for a term can often lead to cancelations or simplification, leading to a more manageable integral.
  • 😑 After performing the substitution, the final result can be obtained by integrating the simplified expression and undoing the substitution.
  • ❓ Understanding how to simplify and integrate complex polynomials is an important skill in calculus.
  • 😫 The process of u substitution involves setting u equal to a term and adjusting the expression accordingly.
  • 😫 This u substitution technique is a form of substitution, but not the traditional u substitution where we set u equal to the derivative of a term.

Transcript

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Questions & Answers

Q: How can we simplify the integral of a complex polynomial?

One approach is to make a substitution, like setting u=x-1, to simplify the expression and make it easier to integrate.

Q: What is the advantage of using a substitution in integration?

A substitution allows us to transform the integral into a simpler form, making the integration process more straightforward and manageable.

Q: How did the substitution u=x-1 simplify the expression?

By substituting u for x-1, the polynomial x-1 to the fifth becomes u to the fifth, simplifying the expression and making it easier to integrate.

Q: How did the expression change after the substitution?

The expression changed to (u+4)u^5 du, where u represents x-1. This form is simpler and more conducive to integration.

Summary & Key Takeaways

  • Instead of expanding a complex polynomial, we can simplify it by making a substitution, which makes it easier to integrate.

  • By setting u=x-1, we can rewrite the expression as a simpler form: (u+4)u^5 du.

  • After distributing and integrating, the final result is (x-1)^7/7 + 2/3(x-1)^6 + C.

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