Students who can differentiate anything often stall on integration. The reason is structural: differentiation follows rules that always apply, while integration requires you to recognise which of several techniques the expression will yield to.
A decision order that works
- Is it a standard result you already know? Take it and stop.
- Can it be rewritten first — expanded, split into separate terms, or written with indices? Most 'hard' integrals become easy here.
- Is there a function and something close to its derivative present? That signals substitution.
- Is it a product of two unrelated functions? That signals integration by parts.
- Is it an algebraic fraction with a factorisable denominator? That signals partial fractions.
- Only then consider a trigonometric identity to reshape it.
Build recognition, not just execution
Practise a mixed set where you only write down which technique you would use and why — no working out. Twenty of those in ten minutes builds the selection skill far faster than five full solutions, and selection is what the exam tests.
Common mark-losers
- Forgetting the constant of integration on an indefinite integral.
- Not changing the limits when substituting in a definite integral.
- Dropping a sign when integrating trigonometric functions.
- Choosing the wrong part to differentiate in integration by parts, making it worse rather than simpler.
- Treating area below the axis as negative when the question asks for total area.
Check your answer for free
Differentiate what you got. If it does not return the original expression, you know immediately, before the marks are gone. It takes thirty seconds and is the single most underused habit in the whole topic.