Applying the substitution method:

• The composite function f(g(x))</latex> must have $g'(x)$ present in the integrand.


  True
• What is the substitution for the integral $\int \sin(5x) \, dx$?


  $u =$$5x$
• When finding $du$, you differentiate $u$ with respect to x
• Match the aspect with its substituted expression:

  Original Variable ↔️ $x$

  Substituted Function ↔️ $f(u)$

  Substituted Derivative ↔️ $du$
• When applying the substitution method, the first crucial step is to identify the appropriate part of the integrand
• In the substituted integrand, $g(x)$ is replaced by $u$


  True
• In the example $\int \sin(5x) \, dx$, the function to substitute is $u =$$5x$


  True
• The derivative $du$ is found by differentiating $u$ with respect to $x$


  True
• In the example $\int x e^{x^{2}} \, dx$, the choice of $u$ is $x^{2}$


  True
• When rewriting the integral, the differential $dx$ is replaced by du
• In the example $\int \sin(5x) \, dx$, the substituted integral is \frac{1}{5} \int \sin(u) \, du
• What is the next step after identifying the appropriate part of the integrand to substitute and choosing the new variable $u$ and its differential $du$?


  Rewrite the original integral
• In the substitution method, the original variable $x$ is replaced with the new variable \blanku
• After solving the new integral, the variable $u$ must be replaced with its original expression in terms of \blankx
• When applying the substitution method, it is necessary to find a function $u$ and its derivative $du$ within the integrand


  True
• The derivative $du$ of $u =$$g(x)$ is given by du = g'(x) \, \blankdx
• To choose a suitable $u$, look for a composite function in the integrand.


  True
• The term $g'(x) \, dx$ must be present or derivable from the original integrand to apply the substitution method.


  True
• What replaces the derivative $g'(x) \, dx$ in the substituted integral?


  $du$
• After solving the new integral in terms of $u$, substitute $u$ back with g(x).
• The constant of integration $+$$C$ is added because the derivative of a constant is zero.


  True
• What is the final expression of the integral after substituting back $u$ with $g(x)$?


  $F(g(x)) +$$C$
• What is the result of integrating $\frac{1}{5} \int \sin(u) \, du$?


  $- \frac{1}{5} \cos(u) +$$C$
• Why is it important to substitute back the original variable $x$ after solving the integral in terms of $u$?


  Express in original variable
• Adding $+$$C$ to an indefinite integral represents all possible antiderivatives
• In the substitution method, we identify a function $u$ and its derivative du
• For the integral $\int (2x + 3)^{4} \, dx$, the derivative $du$ is $2 \, dx$
• Match the aspect of the integrand with its substitution:

  Function to Substitute ↔️ $g(x)$

  Derivative to Use ↔️ $g'(x) \, dx$

  Goal ↔️ Simplify the integral
• The derivative du</latex> must be present or derivable from the original integrand.


  True
• The substitution method involves finding a function $u$ and its derivative $du$ within the integrand.


  True
• Why should choosing the right substitution simplify the integral?

  To make evaluation easier
• What is the derivative of $u =$$2x +$$3$?


  $du =$$2 \, dx$
• Steps to choosing a suitable $u$ for substitution


  1️⃣ Look for composite functions $f(g(x))$

  2️⃣ Choose $u =$$g(x)$

  3️⃣ Find the derivative $du =$$g'(x) \, dx$

  4️⃣ Ensure $g'(x) \, dx$ is derivable from the integrand
• What is the derivative of $u =$$3x +$$5$?


  du = 3 \, dx</latex>
• The derivative du</latex> must always be present in or derivable from the original integrand


  True
• In the example \int (2x + 3)^{4} \, dx</latex>, after substitution, the new integral is $\frac{1}{2} \int u^{4} \, du$


  True
• If u = 5x</latex>, then <latex>du = 5 \, \blankdx
• Match the step of the substitution method with its description:

  Solving the New Integral ↔️ Integrate $f(u) \, du$ to find the indefinite integral in terms of $u$

  Substituting Back ↔️ Replace $u$ with its original expression $g(x)$

  Adding the Constant ↔️ Include $+$$C$ to represent the constant of integration
• What is the final integral of $\int \sin(5x) \, dx$ after substitution and evaluation?


  $- \frac{1}{5} \cos(5x) +$$C$
• Match the aspect of the substitution method with its description:

  Original Integrand ↔️ $\int f(g(x)) \, dx$

  Substituted Integrand ↔️ $\int f(u) \, du$