Integrals of logarithmic functions (the 1/x rule)

The rule

• ∫ dx/x = ln|x| + C. The modulus is compulsory: for x < 0, d/dx ln(−x) = (−1)/(−x) = 1/x, so ln|x| is an antiderivative on both sides of zero.
• Linear argument: ∫ dx/(ax + b) = ln|ax + b|/a + C.
• Numerator equal to the derivative of the denominator: ∫ f′(x)/f(x) dx = ln|f(x)| + C.
• Multiples and powers: ∫ k·dx/x = k ln|x| + C = ln|x|^k + C, and ∫ dx/x² = −1/x + C (a power, not a logarithm).

Worked examples

1. ∫ dx/(3x − 5) = ln|3x − 5|/3 + C.
2. ∫ (2x + 1)/(x² + x + 4) dx: numerator is the derivative of the denominator → ln(x² + x + 4) + C (the quadratic is always positive, so the modulus may be dropped).
3. ∫ dx/x³ = ∫ x⁻³ dx = x⁻²/(−2) + C = −1/(2x²) + C. Do not use the log rule unless the power of x in the denominator is exactly one.
4. ∫ tan x dx = ∫ sin x/cos x dx = −ln|cos x| + C = ln|sec x| + C.
5. ∫ ln x dx = x ln x − x + C (by parts, section 08).