Applying the substitution method:

• Substitute $u$ and du</latex> into the integral
• The substitution method requires choosing a $u$ whose derivative is also present in the integral.


  True
• To calculate $du$, you must take the derivative of $u$ with respect to $dx$.


  True
• Match the function type with its $u$ and $du$:


  Power Functions ↔️ $u =$$x^{2}$, $du =$$2x \, dx$

  Trigonometric Functions ↔️ $u =$$2x +$$3$, $du =$$2 \, dx$

  Logarithmic Functions ↔️ $u =$$\ln x$, $du =$$\frac{1}{x} \, dx$

  Exponential Functions ↔️ $u =$$5x - 2$, $du =$$5 \, dx$
• When calculating $du$, you must take the derivative
• Steps to perform substitution in an integral

  1️⃣ Identify $u$ and $du$

  2️⃣ Substitute $u$ and $du$ into the integral

  3️⃣ Simplify the new integral
• For exponential functions, the exponent is typically chosen as $u$.


  True
• After calculating $du$, both $u$ and $du$ are substituted into the original integral.


  True
• What is the next step after identifying the appropriate $u$ variable and calculating $du$?


  Substitute u and du
• Substituting $u$ and $du$ into the original integral simplifies it for evaluation.


  True
• Replace the original differential (e.g. $dx$) with the du
• For the integral \int 2x e^{x^{2}} \, dx</latex>, the correct substitutions are $u =$$x^{2}$ and $du =$$2x \, dx$.


  True
• After substituting $u$ and du</latex>, the resulting integral is typically simpler and can be solved using basic integration formulas
• After solving the new integral in terms of $u$, you must substitute back the original variable $x$ for $u$ to express the result in terms of $x$.


  True
• In the substitution method, identifying the $u$ involves selecting a part of the integrand such that its derivative is also present in the integral.
• For the integral \int x\sqrt{x^{2} + 1} \, dx, the correct choice for $u$ is $x^{2} +$$1$.


  True
• What is the next step after identifying the appropriate $u$ variable in the substitution method?


  Calculate $du$
• What is the next step after calculating $du$ in the substitution method?


  Substitute $u$ and $du$
• What must be done after solving the new integral in terms of $u$?


  Substitute back $x$
• What is the purpose of changing the limits of integration when evaluating a definite integral using substitution?

  Match the variable $u$
• What is the first step in the substitution method for integration?

  Identify the $u$
• Steps of the substitution method for definite integrals

  1️⃣ Solve the new integral in terms of $u$

  2️⃣ Substitute back $x$ for $u$

  3️⃣ Evaluate the definite integral
• Match the substitution strategy with its example:

  Power Functions ↔️ \int x\sqrt{x^{2} + 1} \, dx, $u =$$x^{2} +$$1$

  Trigonometric Functions ↔️ $\int \cos(2x + 3) \, dx$, $u =$$2x +$$3$

  Logarithmic Functions ↔️ $\int \frac{\ln x}{x} \, dx$, $u =$$\ln x$

  Exponential Functions ↔️ $\int e^{5x - 2} \, dx$, $u =$$5x - 2$
• For the integral $\int 2x e^{x^{2}} \, dx$, choose u = x^{2}</latex>, then $du =$$2x \, dx$, and the integral simplifies to $\int e^{u} \, du$, which has the solution e^{u}
• If $u =$$x^{2}$, then du = 2x \, dx</latex>, which is expressed in terms of the original differential dx
• The key to expressing $du$ is in terms of $dx$ to simplify the integral.


  True
• After calculating $du$, you must substitute both $u$ and $du$ into the original integral
• The correct choice of $u$ is essential for simplifying integrals in the substitution method.


  True
• The first step in substituting $u$ and $du$ is to identify the expressions from the previous steps</latex>
• After identifying the appropriate u variable and calculating du, the next step is to substitute both u and du
• What should you replace occurrences of the original variable (e.g. $x$) with when substituting $u$?


  u expression
• What is the final step after substituting $u$ and $du$ into the integral?


  Simplify the new integral
• Steps for substituting $u$ and $du$ in the integral $\int 2x e^{x^{2}} \, dx$


  1️⃣ Replace $x^{2}$ with $u$: $\int 2x e^{u} \, dx$

  2️⃣ Replace $dx$ with $(du / 2x)$: $\int e^{u} \, (du / 2x)$

  3️⃣ Simplify: $\int e^{u} \, du$
• What is the simplified integral of $\int e^{u} \, du$?


  $e^{u} +$$C$
• What is the final result of the integral $\int 2x e^{x^{2}} \, dx$ after substituting back $x$?


  $e^{x^{2}} +$$C$
• Match the substitution strategy with its guideline:

  Composite Functions ↔️ Choose the inner function as $u$

  Exponents ↔️ Choose the exponent as $u$
• What does calculating $du$ involve?


  Taking the derivative of u</latex>
• If $u =$$x^{2}$, then du = 2x \, dx
• The key is to express $du$ in terms of the original differential dx
• After substituting $u$ and $du$, the resulting integral is typically simpler and can be solved using basic integration formulas.


  True