Integrals of Exponential and Logarithmic Functions: Step-by-Step Guide
The exponential function e^x is its own antiderivative, and 1/x integrates to ln|x|. More generally, e^(kx) integrates to (1/k)e^(kx), a^x integrates to a^x / ln(a), and integrals involving ln(x) typically require integration by parts.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Identify the form. Decide whether the integrand is purely exponential, purely 1/x, a logarithm, or a product. This determines whether you use a basic rule, substitution, or integration by parts.\[\int e^{kx}\,dx,\; \int \frac{1}{x}\,dx,\; \int \ln x\,dx\]
- Apply the matching rule. Use the standard antiderivatives: e^x integrates to e^x, e^(kx) integrates to (1/k)e^(kx), 1/x integrates to ln|x|, and a^x integrates to a^x/ln(a).\[\int e^{kx}\,dx = \frac{1}{k}e^{kx}+C,\;\; \int \frac{dx}{x} = \ln|x|+C\]
- Use substitution if the exponent is composite. If the exponent is a function g(x) and g'(x) is present, let u = g(x) so du captures the derivative.\[\int e^{g(x)}\,g'(x)\,dx = e^{g(x)} + C\]
- Use parts for ln(x) factors. For integrals like integral of ln(x) dx or x ln(x) dx, set u = ln(x) and dv equal to the remaining factor, then apply the by-parts formula.\[\int u\,dv = uv - \int v\,du\]
- Add +C and verify. Include the constant of integration and differentiate to check.\[\frac{d}{dx}[F(x)+C] = f(x)\]
Worked example
Evaluate the integral of e^(3x) dx.
Step 1: The integrand is exponential of the form e^(kx) with k = 3.
Step 2: Apply the rule: integral of e^(kx) dx = (1/k) e^(kx) + C.
Step 3: Substitute k = 3 to get (1/3) e^(3x) + C.
Step 4: Verify by differentiating: d/dx[(1/3)e^(3x)] = (1/3)(3)e^(3x) = e^(3x). Matches.
Final answer: (1/3) e^(3x) + C.
Tips and common mistakes
- Memorize the four core rules: e^x to e^x, e^(kx) to (1/k)e^(kx), 1/x to ln|x|, a^x to a^x/ln(a).
- Always include the absolute value in ln|x|; the integrand 1/x is defined for negative x too.
- For e^(g(x)), look for g'(x) in the integrand and use u = g(x) substitution.
- Integral of ln(x) dx is x ln(x) - x + C. Memorize this since it comes up constantly.
- When in doubt with ln(x), use integration by parts with u = ln(x), dv = dx.
- For a^x with a not equal to e, write a^x = e^(x ln a) if you find that easier than memorizing a^x / ln(a).
- Always differentiate your final answer to confirm it matches the integrand.
- Don't forget the constant +C on indefinite integrals.