1.8 Integration

• The constant of integration, C, is included because the derivative of any constant is zero.


  True
• What is integration in mathematical terms?

  Reverse of differentiation
• What is the integral of x³ using the Power Rule for Integration?

  $\frac{x^{4}}{4} +$$C$
• What is the new exponent of x³ after applying the Power Rule for Integration?

  4
• How do you find the integral of $sin(x)$?


  cos(x) + C</latex>
• When integrating, the constant of integration is denoted by C
• Steps for integration by substitution:

  1️⃣ Identify the substitution variable u

  2️⃣ Differentiate u to get du

  3️⃣ Substitute u and du into the integral

  4️⃣ Evaluate the integral with respect to u

  5️⃣ Substitute back the original variable x
• Match the step with the corresponding action in integration by substitution:

  Identify u ↔️ $u =$$x^{2}$

  Find du ↔️ $du =$$2x dx$

  Substitute ↔️ $∫ e^{u} du$

  Evaluate ↔️ $e^{u} +$$C$
• What does dv represent in the integration by parts formula?

  The differential of the other function
• If u = x, then du = dx.

  True
• The Power Rule for Integration states that for any constant n not equal to -1, $∫ x^{n} dx =$$\frac{x^{n + 1}}{n + 1} +$$C$.
• The notation for finding the derivative is d/dx, while the notation for integration is $∫$.
• The integral of $sin(x)$ is - cos(x) + C, while the integral of $cos(x)$ is $sin(x)$ + C.
• What type of function is Integration by Substitution used for?

  Composite functions
• What is the integral of $sin(x)$?


  $- cos(x) +$$C$
• What is the purpose of finding du in integration by substitution?

  Express the integral in terms of u
• What is the key benefit of integration by substitution?

  Transforms the integral into a simpler form
• Steps for integration by parts:

  1️⃣ Identify the functions u and dv

  2️⃣ Differentiate u to get du

  3️⃣ Integrate dv to get v

  4️⃣ Substitute into the formula
• What is the value of v if dv = $e^{x} dx$?


  $e^{x}$
• If dv = e^x dx, then v = e^x
• Steps to apply Integration by Parts

  1️⃣ Identify the functions u and dv in the original integral.

  2️⃣ Differentiate u to get du and integrate dv to get v.

  3️⃣ Substitute these values into the formula to evaluate the integral.
• Integration is the reverse process of differentiation
• What does the Power Rule for Integration state for n ≠ -1?

  $∫ x^{n} dx =$$\frac{x^{n + 1}}{n + 1} +$$C$
• What is the integral of $sin(x)$?


  $- cos(x) +$$C$
• The integral of $sec(x)$ is ln(sec(x)) + C</latex>
• The final step in Integration by Substitution is to substitute back the original variable x.


  True
• The formula for integration by parts is ∫ u dv = uv - ∫ v du
• Definite integrals produce a numerical value, unlike indefinite integrals.

  True
• The linearity property of definite integrals states that ∫_{a}^{b} [cf(x) + dg(x)] dx = c ∫_{a}^{b} f(x) dx + d ∫_{a}^{b} g(x) dx</latex>.linearity
• The integral of an odd function over a symmetric interval is always 0.


  True
• What are the key steps for applying the Power Rule for Integration?

  Increase exponent, divide
• The integral of $sec(x)$ is $ln(sec(x)) +$$C$.


  True
• The power rule is applied to trigonometric functions by treating them as $x^{n}$.


  True
• In integration by substitution, the final step is to substitute back the original variable x
• The formula for integration by parts is $∫ u dv =$$uv - ∫ v du$.


  True
• In integration by parts, $∫ u dv$ is equal to $uv - ∫ v$du
• The final integral after applying integration by parts can often be solved using the power rule.


  True
• What is the formula for Integration by Parts?

  $∫ u dv =$$uv - ∫ v du$
• What is the value of du if u = x?

  dx
• What is the notation for integration?

  $∫ f(x) dx$