Explanation
Differentiation finds the rate of change; integration reverses it. For polynomials: bring down the power and reduce by one when differentiating; add one to the power and divide by the new power when integrating (and add +c for indefinite integrals). Chain rule, product rule, and substitution extend this to more complex functions. Always state the rule you’re using and show one step at a time so the mark scheme can award method marks.
Example question
Find the exact value of ∫₁^e (ln x)/x dx.
Worked solution
Use substitution: u = ln x ⇒ du/dx = 1/x ⇒ dx = x du. When x = 1, u = 0; when x = e, u = 1. ∫ (ln x)/x dx = ∫ u du = u²/2 + c ⇒ definite integral = [u²/2]₀¹ = 1/2 − 0 = 1/2.
Exam tips
- • Always write +c for indefinite integrals.
- • When using substitution, change limits to the new variable so you don’t need to substitute back.
- • If the question says "exact", leave answers in terms of π, e, or surds; don’t round.