Answer:
[tex]\displaystyle \int {e^{5x}} \, dx = \boxed{ \frac{e^{5x}}{5} + C }[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Methods: U-Substitution
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \int {e^{5x}} \, dx[/tex]
Step 2: Integrate Pt. 1
Identify variables for u-substitution.
- Set u:
[tex]\displaystyle u = 5x[/tex] - [u] Differentiate [Derivative Rules and Properties]:
[tex]\displaystyle du = 5 \ dx[/tex]
Step 3: Integrate Pt. 2
- [Integral] Rewrite [Integration Property - Multiplied Constant]:
[tex]\displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\\end{aligned}[/tex] - [Integral] Apply Integration Method [U-Substitution]:
[tex]\displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\\end{aligned}[/tex] - [Integral] Apply Exponential Integration:
[tex]\displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\& = \frac{e^u}{5} + C \\\end{aligned}[/tex] - [u] Back-substitute:
[tex]\displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\& = \frac{e^u}{5} + C \\& = \boxed{ \frac{e^{5x}}{5} + C } \\\end{aligned}[/tex]
∴ we have used substitution to evaluate the indefinite integral.
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
