Calculus Substitution Method Quiz
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Questions and Answers

What is the correct substitution used to evaluate the integral of $ rac{1}{1+e^x}$?

  • Let $t = e^x + 1$ (correct)
  • Let $t = e^{-x} + 1$
  • Let $t = e^x - 1$ (correct)
  • Let $t = 1 + rac{1}{e^x}$
  • The primitive of $ae^x + be^{-x}$ is derived by which substitution?

  • $ae^x + b = t$
  • $e^x - b = t$
  • $ae^x + be^{-x} = t$
  • $ae^x - be^{-x} = t$ (correct)
  • After making the substitution $2 + 3logx = t$, what does the integral transform into?

  • $ rac{3}{x}dx$
  • $ rac{1}{t}dt$
  • $ rac{dt}{t}$ (correct)
  • $dt + 3$
  • In the evaluation of $ rac{1}{ ext{sqrt}(x+x)}$, which relationship is used for substitution?

    <p>$1 + ext{sqrt}(x) = t$</p> Signup and view all the answers

    What advanced technique might be needed to evaluate the primitive of $ rac{2e^x - 3}{4e^x + 1}$?

    <p>Trigonometric substitution</p> Signup and view all the answers

    Study Notes

    Evaluating Primitives with Substitution

    • Substitution: A method to simplify integrals by replacing parts of the integrand with new variables.
    • Goal of Substitution: To transform the integral into a simpler form that is easier to integrate.
    • Steps Involved:
      • Choose a Substitution: Select a part of the integrand to replace with a new variable.
      • Differentiate: Differentiate the chosen substitution to find the differential of the new variable.
      • Transform: Substitute the chosen variable and its differential into the integral, transforming it into a simpler form.
      • Solve: Integrate the transformed integral.
      • Substitute Back: Replace the new variable with its original expression to get the final answer.

    Examples of Substitution

    • Example 1: ∫(1/(1+e^x)) dx

      • Substitution: e^x + 1 = t
      • Differential: e^x dx = dt
      • Transform: ∫(-dt/t)
      • Solution: -log|t| + c
      • Substitute Back: -log|e^x + 1| + c
    • Example 2: ∫(ae^x + be^{-x}) dx

      • Substitution: ae^x - be^{-x} = t
      • Differential: (ae^x + be^{-x}) dx = dt
      • Transform: ∫(dt/t)
      • Solution: log|t| + c
      • Substitute Back: log|ae^x - be^{-x}| + c
    • Example 3: ∫(1/(2x + 3logx)) dx

      • Substitution: 2 + 3logx = t
      • Differential: (3/x) dx = dt, (dx/x) = dt/3
      • Transform: ∫(dt/t)
      • Solution: log|t| + c
      • Substitute Back: log|2 + 3logx| + c
    • Example 4:∫(1/√(x+x)) dx

      • Substitution: 1 + √x = t
      • Differential: (1/(2√x))dx = dt, dx/√x = 2dt
      • Transform: ∫(2dt/t)
      • Solution: 2log|t| + c
      • Substitute Back: 2log|1 +√x| + c
    • Example 5: ∫(2e^x-3)/(4e^x+1) dx

      • Substitution: This example might require more advanced integration methods or techniques, such as trigonometric substitution.

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    Description

    Test your understanding of the substitution method for evaluating integrals. This quiz covers the steps involved in substitution, from choosing a variable to transforming and solving the integral. Practice with examples to solidify your knowledge of this essential calculus technique.

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