6.9 Integrating Using Substitution

• Substitution is a technique used in integration to simplify integrals involving composite functions by reversing the chain rule.
• The first step in substitution is to choose a suitable composite part of the function as u.
• Steps to integrate using substitution

  1️⃣ Choose $u$

  2️⃣ Compute $du$

  3️⃣ Transform the integral

  4️⃣ Integrate with respect to $u$

  5️⃣ Substitute back to $x$
• After integrating with respect to $u$, you must substitute back to the original variable x.
• What is the result of integrating $\int (3x + 2)^{5} dx$ using substitution?


  $\frac{1}{18} (3x + 2)^{6} +$$C$
• To identify suitable substitutions, look for composite functions where the derivative of the inner function is present in the integral.
• For $\int (3x + 2)^{5} dx$, a suitable substitution is $u =$$3x +$$2$ because its derivative is a constant.
• What is the first step in performing a substitution in integration?

  Select $u$
• After selecting $u$, the next step is to compute du.
• What should you look for when identifying suitable substitutions in an integral?

  Composite functions
• For the integral $\int (3x + 2)^{5} dx$, u = 3x + 2 is a suitable substitution because its derivative (3) is a constant factor.
• Steps to perform substitution in integration

  1️⃣ Select u

  2️⃣ Compute du

  3️⃣ Transform the integral

  4️⃣ Integrate with respect to u

  5️⃣ Substitute back
• What is the first step in performing substitution in integration?

  Select u
• After transforming the integral, you must integrate with respect to u
• What is the final result of $\int x^{2} e^{x^{3}} dx$ using substitution?


  $\frac{1}{3} e^{x^{3}} +$$C$
• The constant of integration, C, must be added to complete an indefinite integral after substitution.
• What is the purpose of substitution in integration?

  Simplify complex integrals
• Steps to define and use substitution in integration

  1️⃣ Choose u

  2️⃣ Compute du

  3️⃣ Transform the integral

  4️⃣ Integrate

  5️⃣ Substitute back
• For the integral $\int (3x + 2)^{5} dx$, the substitution is $u =$$3x +$$2$, and du = 3 dx</latex>, so $dx =$$\frac{1}{3} du$.3
• What is the transformed integral after substitution in $\int (3x + 2)^{5} dx$?


  $\int u^{5} \frac{1}{3} du$
• Substitution in integration involves reversing the chain rule.
• What should you look for in the integrand to identify composite functions for substitution?

  Inner function derivative
• For the integral \int \sin(2x) dx</latex>, the substitution is $u =$$2x$, and $du =$$2 dx$, so $dx =$$\frac{1}{2} du$.2
• What is the transformed integral after substitution in $\int \sin(2x) dx$?


  $\frac{1}{2} \int \sin(u) du$
• The inner function of a composite function is chosen as $u$ for substitution.
• What is the $du$ for the substitution $u =$$x^{2} +$$1$?


  $du =$$2x dx$
• For the integral $\int (x^{2} +$$1)^{3} \cdot 2x dx$, after substitution, the integral becomes $\int u^{3} du$, which simplifies to $\frac{1}{4} u^{4} +$$C$.\frac{1}{4}
• After substitution, the transformed integral must be integrated with respect to $u$.
• What is the first step in performing the substitution method for integration?

  Identify a composite function
• In the substitution method, $u$ is chosen as the inner function of the composite function.
• Steps to compute $du$ and solve for $dx$


  1️⃣ Compute $du$ with respect to $x$

  2️⃣ Rearrange the equation to solve for $dx$
• After substituting $u$ and $du$, the integral is transformed into a simpler form in terms of $u$.
• What is the final step in the substitution method for integration?

  Replace $u$ with its original expression
• In the example, $u$ is identified as x^{2} + 1
• Match the action with the corresponding step in the substitution method:

  Identify $u$ ↔️ $u =$$x^{2} +$$1$

  Compute $du$ ↔️ $du =$$2x dx$

  Integrate ↔️ $\int u^{3} du =$$\frac{1}{4} u^{4} +$$C$
• What is the standard integration rule for $\int \frac{1}{u} du$?


  $\ln |u| +$$C$
• The constant of integration, $C$, must always be added after integration.
• Why is it necessary to substitute u</latex> back into its original expression in terms of $x$?


  To express the result in terms of $x$
• Steps to evaluate definite integrals using substitution

  1️⃣ Choose $u$ and compute $du$

  2️⃣ Change limits of integration

  3️⃣ Transform the definite integral

  4️⃣ Evaluate the transformed integral
• How are the new limits of integration found when evaluating definite integrals using substitution?

  Substitute original limits into $u =$$f(x)$