Question

Evaluate the following: integral S (e^3x)/((e^6x)+1) dx
(hint: u=e^3x)

Answer 1

`\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (arctan(e^(3x)))/(3) + C`

General Formulas and Concepts:

Algebra I

• Exponential Rule [Powering]:
  `\displaystyle (b^m)^n = b^(m \cdot n)`

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = e^(3x)`
2. [u] Differentiate [Exponential Differentiation, Chain Rule]:
   `\displaystyle du = 3e^(3x) \ dx`

Step 3: Integrate Pt. 2

1. [Integrand] Rewrite [Exponential Rule - Powering]:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = \int {(e^(3x))/((e^(3x))^2 + 1)} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (1)/(3)\int {(3e^(3x))/((e^(3x))^2 + 1)} \, dx`
3. [Integral] U-Substitution:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (1)/(3)\int {(1)/(u^2 + 1)} \, du`
4. [Integral] Arctrig Integration:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (1)/(3) \bigg[ (1)/(1)arctan \Big( (u)/(1) \Big) \bigg] + C`
5. Simplify:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (arctan(u))/(3) + C`
6. Back-Substitute:
   `\displaystyle \int {(e^(3x))/(e^(6x) + 1)} \, dx = (arctan(e^(3x)))/(3) + C`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e