Exponential functions are central in calculus because they model quantities that grow or decay at rates proportional to their current size. Integration lets us find accumulated change, such as total growth, total charge, or total population over time. For y = e^x, the antiderivative has the same shape as the original function, making it one of the simplest and most important integration rules.
Shading the area under an exponential curve helps connect the formula to the idea of accumulation.
Key Facts
- Integral of e^x: ∫ e^x dx = e^x + C
- Integral of e^(kx): ∫ e^(kx) dx = (1/k)e^(kx) + C, where k ≠ 0
- Integral of a^x: ∫ a^x dx = a^x / ln(a) + C, where a > 0 and a ≠ 1
- Substitution rule: if u = g(x), then ∫ e^(g(x))g'(x) dx = e^(g(x)) + C
- Definite integral: ∫ from a to b e^x dx = e^b - e^a
- Integration reverses differentiation: if F'(x) = f(x), then ∫ f(x) dx = F(x) + C
Vocabulary
- Antiderivative
- An antiderivative of f(x) is a function F(x) whose derivative is f(x).
- Exponential function
- An exponential function has the variable in the exponent, such as e^x or 2^x.
- Natural base e
- The number e is a special constant about 2.718 that makes d/dx e^x = e^x.
- Constant of integration
- The constant C represents all vertical shifts of an antiderivative in an indefinite integral.
- Substitution
- Substitution is a method for simplifying an integral by replacing a complicated expression with a new variable.
Common Mistakes to Avoid
- Forgetting the factor 1/k in ∫ e^(kx) dx is wrong because differentiating e^(kx) gives k e^(kx), so the antiderivative must include division by k.
- Writing ∫ a^x dx = a^x + C is wrong for bases other than e because the derivative of a^x is a^x ln(a).
- Dropping the constant C in an indefinite integral is wrong because antiderivatives differ by any constant vertical shift.
- Using substitution without changing du correctly is wrong because the integral only simplifies when the derivative of the inside function is accounted for.
Practice Questions
- 1 Evaluate the indefinite integral ∫ 5e^(2x) dx.
- 2 Compute the definite integral ∫ from 0 to 3 e^(0.5x) dx.
- 3 Explain why ∫ e^(x^2) dx cannot be solved using the simple rule ∫ e^(g(x))g'(x) dx = e^(g(x)) + C.