6.9 Integrating Using Substitution

• The substitution method is useful when the original integral cannot be easily evaluated
• Steps to evaluate ∫ (2x + 3)^5 dx using substitution

  1️⃣ Let u = 2x + 3

  2️⃣ Differentiate: du = 2 dx, so dx = du / 2

  3️⃣ Rewrite the integral: ∫ u^5 (du / 2) = (1/2) ∫ u^5 du

  4️⃣ Evaluate: (1/2) * (u^6 / 6) + C = u^6 / 12 + C

  5️⃣ Substitute back: (2x + 3)^6 / 12 + C
• Steps to evaluate ∫ (2x + 3)^5 dx using the substitution method

  1️⃣ Let u = 2x + 3

  2️⃣ Find dx in terms of u and du: du/dx = 2, so dx = du/2

  3️⃣ Rewrite the integral: ∫ u^5 (du/2)

  4️⃣ Evaluate the new integral: (1/2) ∫ u^5 du = (1/12) u^6 + C

  5️⃣ Substitute back: (2x + 3)^6 / 12 + C
• To find the differential du, you must differentiate u with respect to x
• The substitution method allows expressing the original differential dx in terms of du.
• Incorrectly expressing du can lead to an incorrect integral.

  True
• When substituting into the original integral, dx is replaced with its equivalent expression in terms of du.
• Steps for using the substitution method

  1️⃣ Choose a suitable expression for u within the integral

  2️⃣ Find du/dx and rearrange to express dx in terms of du

  3️⃣ Substitute u and dx into the integral

  4️⃣ Evaluate the new integral

  5️⃣ Substitute back to express the result in terms of x
• What strategy is recommended for choosing u in a composite function?

  Inner function
• What is the first step in the general process of the substitution method?

  Choose the substitution variable u
• What happens if du is not found accurately in the substitution method?

  The integral will be incorrect
• After integrating the transformed integral in terms of u, substitute back the original variable x
• The substitution method is a technique used to evaluate integrals by performing a change of variable
• Choosing the appropriate substitution variable u is crucial for simplifying integrals.


  True
• The substitution method simplifies integrals by replacing parts of the original expression with a new variable u.

  True
• The goal of choosing the appropriate substitution is to simplify the expression and make it easier to integrate
• What two variables are essential for the substitution method in integration?

  u and du
• Steps for using the substitution method to find du

  1️⃣ Identify the substitution variable u

  2️⃣ Differentiate u with respect to x to find du/dx

  3️⃣ Rearrange to solve for dx in terms of du
• The substitution method relies on finding the differential du of the substitution variable u.
• For definite integrals, the limits of integration must be converted from x to u.


  True
• After substitution, the new integral is evaluated in terms of u.
• When identifying derivative pairs, look for a function and its derivative within the integrand.
• After choosing the substitution variable u, the next step is to differentiate u with respect to x
• When adjusting the limits of integration for a definite integral, the limits must be converted from x to u
• Adding the constant of integration C ensures the completeness of the integral's solution


  True
• Steps in the substitution method for evaluating integrals

  1️⃣ Identify a variable in the integrand that can be replaced by a new variable, u

  2️⃣ Express the original differential, dx, in terms of the new variable u and its differential, du

  3️⃣ Rewrite the integral in terms of u and du

  4️⃣ Evaluate the new integral in terms of u, then substitute back the original variable
• Match the substitution strategy with its description:

  Inner Functions ↔️ Choose the inner function within a composite function

  Derivative Pairs ↔️ Look for function-derivative pairs in the integrand

  Powers & Logarithms ↔️ Use the power or logarithmic function as u
• To find dx in terms of u and du, you must differentiate u with respect to x
• Steps to find the differential du in the substitution method

  1️⃣ Identify the substitution variable u

  2️⃣ Differentiate u with respect to x to find du/dx

  3️⃣ Rearrange to solve for dx in terms of du
• Correctly finding du is essential for the substitution method to work properly.


  True
• Expressing dx in terms of du simplifies the integral.
• Finding du/dx requires differentiating u with respect to x.


  True
• Why is adjusting the original integral necessary in the substitution method?

  To evaluate in terms of u
• The substitution method is particularly useful for composite functions.


  True
• The goal of substitution is to simplify the integrand and make it easier to integrate.


  True
• The substitution method requires rearranging du/dx to solve for dx in terms of du


  True
• Steps to integrate the transformed integral in the substitution method

  1️⃣ Integrate the expression in terms of u

  2️⃣ Evaluate the integral and obtain the result in terms of u

  3️⃣ Substitute back the original variable x

  4️⃣ Simplify the final expression if necessary
• What should always be added to the end of an indefinite integral?

  C