Some antiderivatives produce powers, but integrals involving a variable expression in the denominator often produce logarithms. The key pattern is that the derivative of ln|u| is 1/u times du/dx. This makes logarithmic integrals essential for solving problems in calculus, physics, chemistry, biology, and engineering.
Recognizing the pattern quickly helps turn a complicated-looking fraction into a simple substitution problem.
The central rule is ∫ 1/u du = ln|u| + C, where u can be a variable or a more complicated expression. When an integral has the form f'(x)/f(x), its antiderivative is ln|f(x)| + C. The absolute value is needed because the input of a real logarithm must be positive, while f(x) may be negative on part of its domain.
Substitution is the main tool for matching an integrand to this logarithmic form.
Key Facts
- Basic logarithm rule: ∫ 1/u du = ln|u| + C.
- Chain rule pattern: ∫ f'(x)/f(x) dx = ln|f(x)| + C.
- If u = g(x), then du = g'(x) dx, so the numerator must match the derivative of the denominator up to a constant factor.
- Example: ∫ 2/(2x + 5) dx = ln|2x + 5| + C.
- Example: ∫ x/(x^2 + 4) dx = 1/2 ln|x^2 + 4| + C.
- Constant multiple adjustment: ∫ 1/(ax + b) dx = 1/a ln|ax + b| + C, for a ≠ 0.
Vocabulary
- Antiderivative
- An antiderivative of a function is another function whose derivative gives the original function.
- Natural logarithm
- The natural logarithm ln x is the logarithm with base e and is the antiderivative pattern connected to 1/x.
- Substitution
- Substitution is a method that replaces a complicated expression with a new variable to simplify an integral.
- Absolute value
- Absolute value gives the nonnegative size of a number and is used in ln|u| so the logarithm receives a positive input.
- Constant of integration
- The constant of integration C represents the family of functions that have the same derivative.
Common Mistakes to Avoid
- Writing ∫ 1/u du = 1/2 u^2 + C. This is wrong because the power rule does not apply when the exponent is -1, and the correct result is ln|u| + C.
- Dropping the absolute value in ln|u|. This is wrong in real-valued calculus because u may be negative, and ln(u) is not defined for negative u.
- Forgetting to adjust for the derivative of the inside expression. This is wrong because ∫ 1/(3x + 2) dx is 1/3 ln|3x + 2| + C, not ln|3x + 2| + C.
- Using a logarithm whenever there is a denominator. This is wrong because the numerator must match the derivative of the denominator up to a constant factor, as in f'(x)/f(x).
Practice Questions
- 1 Evaluate ∫ 5/(5x - 4) dx.
- 2 Evaluate ∫ 6x/(3x^2 + 7) dx.
- 3 Explain why ∫ x/(x^2 - 9) dx results in a logarithm, but ∫ x/(x^3 - 9) dx does not match the simple ln|u| pattern directly.