- Perform integrations on functions that include exponential terms
- Solve integrals that feature logarithmic functions
Integrals of Exponential Functions
The Main Idea
- Exponential functions are their own derivatives and integrals
- Key integration formulas:
- [latex]\int e^x dx = e^x + C[/latex] [latex]\int a^x dx = \frac{a^x}{\ln a} + C[/latex]
- Substitution is often used for more complex exponential integrals
- Exponential functions are common in real-life applications, especially in growth and decay scenarios
- Integration Process:
- For simple exponentials, apply the formula directly
- For complex expressions, use substitution with u as the exponent of [latex]e[/latex]
- Substitution Tips:
- If only one [latex]e[/latex] exists, choose its exponent as [latex]u[/latex]
- If multiple [latex]e[/latex]‘s exist, choose the more complicated function as [latex]u[/latex]
- Common Mistakes:
- Don’t treat exponents on e like polynomial exponents
- Be careful when both exponentials and polynomials are present
Find the antiderivative of the function using substitution: [latex]{x}^{2}{e}^{-2{x}^{3}}.[/latex]
Find the antiderivative of [latex]{e}^{x}{(3{e}^{x}-2)}^{2}.[/latex]
Evaluate the indefinite integral [latex]\displaystyle\int 2{x}^{3}{e}^{{x}^{4}}dx.[/latex]
Evaluate [latex]{\displaystyle\int }_{0}^{2}{e}^{2x}dx.[/latex]
Suppose a population of fruit flies increases at a rate of [latex]g(t)=2{e}^{0.02t},[/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?
Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?
Evaluate the definite integral using substitution: [latex]{\displaystyle\int }_{1}^{2}\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[/latex]
Integrals Involving Logarithmic Functions
The Main Idea
- Integration of reciprocal functions leads to logarithms
- Key integration formulas for logarithmic functions:
- [latex]\int x^{-1} dx = \ln |x| + C[/latex] [latex]\int \ln x dx = x\ln x - x + C[/latex] [latex]\int \log_a x dx = \frac{x}{\ln a}(\ln x - 1) + C[/latex]
- Substitution is often used for more complex logarithmic integrals
- Logarithmic integrals appear in various applications, including entropy and information theory
- Integration Process:
- For simple reciprocal functions, apply the formula directly
- For complex expressions, use substitution or rewrite in terms of natural logarithms
- Substitution Tips:
- Look for expressions that can be rewritten as [latex]u^(-1)[/latex]
- Be prepared to adjust du when necessary
- Common Challenges:
- Remember the absolute value in the logarithm formula
- Be careful with the domain of logarithmic functions
- Pay attention to the base of the logarithm
Find the antiderivative of [latex]\dfrac{1}{x+2}.[/latex]
Find the antiderivative of [latex]{\text{log}}_{3}x.[/latex]